Unit 6 Class 9 Math Solutions – Trigonometry
Unit 6 Class 9 Math Solutions contain complete exercise-wise PDF solutions of the chapter Trigonometry. Students can find the solutions of Exercise 6.1, Exercise 6.2, Exercise 6.3, Exercise 6.4, Exercise 6.5, Exercise 6.6, and Review Exercise 6 on this page.
This unit begins with angles, quadrants, degrees, minutes, seconds, radians, sectors, and arc length. It then introduces trigonometric ratios, exact values, identities, right-angled triangles, and practical questions involving heights and distances.
The uploaded PDFs include complete calculations, diagrams, right triangles, sector figures, identity proofs, and final answers for the Punjab Board Class 9 Mathematics book.
Students can use these solutions for homework, revision, class tests, annual examinations, and board exam preparation.
Solutions of Exercise 6.1 Unit 6 Class 9 Math Notes
Exercise 6.1 of Unit 6 Class 9 Math Notes covers angles, quadrants, co-terminal angles, degree and radian conversions, arc length, sector area, percentages of circles, and the formation of a cone from a circular sector.
Angles in Standard Position
An angle is in standard position when its initial side lies on the positive x-axis and its vertex is at the origin.
Positive angles are measured anticlockwise, while negative angles are measured clockwise.
The four quadrants are separated by the positive and negative x-axis and y-axis.
Quadrants of an Angle
An angle between 0 degrees and 90 degrees lies in Quadrant I.
An angle between 90 degrees and 180 degrees lies in Quadrant II.
An angle between 180 degrees and 270 degrees lies in Quadrant III.
An angle between 270 degrees and 360 degrees lies in Quadrant IV.
Negative or large angles should first be replaced by a positive co-terminal angle between 0 degrees and 360 degrees.
Co-Terminal Angles
Two angles are co-terminal when their terminal sides are the same.
Co-terminal angles differ by a whole-number multiple of 360 degrees.
\(\theta_{\text{co-terminal}}=\theta+360^\circ n,\qquad n\in\mathbb{Z}\)
For example:
\(65^\circ+360^\circ=425^\circ\)
Therefore, 65 degrees and 425 degrees are co-terminal.
Degrees, Minutes, and Seconds
One degree is divided into 60 minutes, and one minute is divided into 60 seconds.
\(1^\circ=60’\)
\(1’=60”\)
This system is used when an angle must be written more accurately than a whole number of degrees.
Converting Decimal Degrees into Degrees, Minutes, and Seconds
Keep the whole-number part as degrees.
Multiply the decimal part by 60 to obtain minutes.
Keep the whole-number part of the result as minutes, and multiply the remaining decimal part by 60 to obtain seconds.
For example, the method can be represented as:
\(D.d^\circ=D^\circ+M’+S”\)
Converting Degrees, Minutes, and Seconds into Decimal Degrees
Use the formula:
\(\text{Decimal degrees}=D+\frac{M}{60}+\frac{S}{3600}\)
For example:
\(65^\circ32’15”=65+\frac{32}{60}+\frac{15}{3600}\)
Each part must be converted into degrees before the values are added.
Converting Degrees into Radians
To convert degrees into radians, multiply by pi and divide by 180.
\(\text{Radians}=\text{Degrees}\times\frac{\pi}{180}\)
For example:
\(36^\circ=36\times\frac{\pi}{180}=\frac{\pi}{5}\)
Converting Radians into Degrees
To convert radians into degrees, multiply by 180 and divide by pi.
\(\text{Degrees}=\text{Radians}\times\frac{180}{\pi}\)
The factor pi cancels when the radian measure is written as a multiple of pi.
Arc Length of a Sector
When the central angle is measured in radians, the arc length is:
\(s=r\theta\)
Here, \(s\) is the arc length, \(r\) is the radius, and \(\theta\) is the central angle in radians.
When the angle is given in degrees, use:
\(s=\frac{\theta}{360^\circ}\times2\pi r\)
Area of a Sector
When the angle is measured in radians, the area of a sector is:
\(A=\frac{1}{2}r^2\theta\)
When the angle is measured in degrees, use:
\(A=\frac{\theta}{360^\circ}\pi r^2\)
Students should check the unit of the angle before selecting the formula.
Percentage of a Circle
The percentage represented by a sector can be found from its central angle.
For an angle in degrees:
\(\text{Percentage}=\frac{\theta}{360^\circ}\times100\)
For an angle in radians:
\(\text{Percentage}=\frac{\theta}{2\pi}\times100\)
For example, a sector with angle \(\frac{\pi}{8}\) represents:
\(\frac{\pi/8}{2\pi}\times100=6.25\%\)
Sector Bent to Form a Cone
When a circular sector is bent to form a cone, the radius of the sector becomes the slant height of the cone.
The arc length of the sector becomes the circumference of the circular base.
\(\text{Arc length of sector}=2\pi r_{\text{base}}\)
This relationship is used to find the radius of the cone after the sector has been joined.
Solutions of Exercise 6.2 Unit 6 Class 9 Math Notes
Exercise 6.2 introduces trigonometric ratios and their signs in different quadrants. Students use exact values, right triangles, reciprocal ratios, and Pythagoras’ theorem to evaluate expressions and find missing ratios.
The Three Basic Trigonometric Ratios
For an acute angle in a right-angled triangle:
\(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\)
\(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)
\(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\)
The opposite and adjacent sides depend on the selected angle, while the hypotenuse is always opposite the right angle.
Reciprocal Trigonometric Ratios
The remaining three trigonometric ratios are reciprocals of sine, cosine, and tangent.
\(\cosec\theta=\frac{1}{\sin\theta}\)
\(\sec\theta=\frac{1}{\cos\theta}\)
\(\cot\theta=\frac{1}{\tan\theta}\)
Signs of Trigonometric Ratios in the Quadrants
In Quadrant I, all six trigonometric ratios are positive.
In Quadrant II, sine and cosecant are positive.
In Quadrant III, tangent and cotangent are positive.
In Quadrant IV, cosine and secant are positive.
The sign of a ratio should be checked before writing the final answer for an angle outside the first quadrant.
Related Angles
Exercise 6.2 includes angles such as:
\(180^\circ-\theta\)
\(180^\circ+\theta\)
\(360^\circ-\theta\)
\(-\theta\)
Students identify the quadrant of each related angle and then determine the sign of its trigonometric ratios.
Exact Values of Special Angles
The exact values of 30 degrees, 45 degrees, and 60 degrees should be memorized.
\(\sin30^\circ=\frac{1}{2},\qquad \cos30^\circ=\frac{\sqrt3}{2},\qquad \tan30^\circ=\frac{1}{\sqrt3}\)
\(\sin45^\circ=\frac{\sqrt2}{2},\qquad \cos45^\circ=\frac{\sqrt2}{2},\qquad \tan45^\circ=1\)
\(\sin60^\circ=\frac{\sqrt3}{2},\qquad \cos60^\circ=\frac{1}{2},\qquad \tan60^\circ=\sqrt3\)
Finding All Six Ratios from a Triangle
Label the opposite, adjacent, and hypotenuse sides with respect to the given angle.
Write sine, cosine, and tangent first.
Then write cosecant, secant, and cotangent by taking reciprocals.
Simplify fractions and rationalize denominators where required.
Finding Remaining Ratios from One Given Ratio
A given ratio may be treated as a pair of side lengths.
For example, if:
\(\sin\theta=\frac{8}{17}\)
take the opposite side as 8 and the hypotenuse as 17.
Use Pythagoras’ theorem to find the adjacent side:
\(a^2+8^2=17^2\)
This gives \(a=15\), after which the remaining five ratios can be written.
Using Pythagoras’ Theorem
The missing side of a right triangle is found from:
\((\text{hypotenuse})^2=(\text{opposite})^2+(\text{adjacent})^2\)
Only the positive square root is used because the side length of a triangle cannot be negative.
Using Fundamental Identities
Exercise 6.2 also uses the identities:
\(\sin^2\theta+\cos^2\theta=1\)
\(1+\tan^2\theta=\sec^2\theta\)
These identities help students find unknown values and simplify trigonometric expressions.
Evaluating Expressions from a Given Ratio
When a value such as \(\sec\theta=\sqrt2\) is given, first find the related basic ratios.
Use the reciprocal relation to find cosine, and then use an identity to find tangent or sine.
After the required ratios are known, substitute them into each expression and simplify.
Solutions of Exercise 6.3 Unit 6 Class 9 Math Notes
Exercise 6.3 focuses on finding the remaining trigonometric ratios and proving trigonometric identities. The solutions use reciprocal relations, quotient relations, Pythagorean identities, algebraic factorization, rationalization, and standard algebraic formulas.
Finding Remaining Ratios
When one ratio is given, draw or imagine a right triangle and assign suitable side lengths.
Use Pythagoras’ theorem to find the missing side.
Then calculate all five remaining ratios.
Since the questions state that the angle lies in the first quadrant, all ratios are positive.
Pythagorean Identities
The three main Pythagorean identities are:
\(\sin^2\theta+\cos^2\theta=1\)
\(1+\tan^2\theta=\sec^2\theta\)
\(1+\cot^2\theta=\cosec^2\theta\)
These identities may be rearranged to replace one squared ratio with an equivalent expression.
Quotient Identities
Tangent and cotangent may be written as quotients of sine and cosine.
\(\tan\theta=\frac{\sin\theta}{\cos\theta}\)
\(\cot\theta=\frac{\cos\theta}{\sin\theta}\)
This conversion is useful when both sides of an identity need to be written using sine and cosine.
Reciprocal Identities
\(\sec\theta=\frac{1}{\cos\theta}\)
\(\cosec\theta=\frac{1}{\sin\theta}\)
\(\cot\theta=\frac{1}{\tan\theta}\)
Replacing reciprocal ratios often allows common factors to be cancelled.
How to Prove a Trigonometric Identity
Begin with the more complicated side, usually the left-hand side.
Apply identities and algebraic steps until it becomes the other side.
Do not begin by assuming that both sides are equal.
Write one justified step at a time and state the identity used where necessary.
Expanding a Square
Some identities use:
\((a+b)^2=a^2+2ab+b^2\)
For example:
\((\sin\theta+\cos\theta)^2\)
expands to:
\(\sin^2\theta+2\sin\theta\cos\theta+\cos^2\theta\)
Using \(\sin^2\theta+\cos^2\theta=1\), the expression becomes:
\(1+2\sin\theta\cos\theta\)
Multiplying by a Conjugate
An expression containing \(1-\sin\theta\) or \(1-\cos\theta\) in a denominator can often be simplified by multiplying by its conjugate.
For example, the conjugate of \(1-\sin\theta\) is \(1+\sin\theta\).
\((1-\sin\theta)(1+\sin\theta)=1-\sin^2\theta=\cos^2\theta\)
Using Algebraic Factorization
Some identities use difference of squares, sum or difference of cubes, and common factors.
\(a^2-b^2=(a-b)(a+b)\)
\(a^3-b^3=(a-b)(a^2+ab+b^2)\)
For example:
\(\sin^3\theta-\cos^3\theta\)
can be factorized before applying \(\sin^2\theta+\cos^2\theta=1\).
Avoiding Invalid Cancellation
Terms connected by addition or subtraction cannot be cancelled directly.
Cancellation is allowed only after the numerator and denominator have been written as products of factors.
This rule is important in identities containing fractions.
Solutions of Exercise 6.4 Unit 6 Class 9 Math Notes
Exercise 6.4 focuses on exact trigonometric values. Students evaluate special-angle ratios without a calculator and simplify expressions containing 30-degree, 45-degree, and 60-degree angles.
Degrees and Radians for Special Angles
The most frequently used equivalent angles are:
\(30^\circ=\frac{\pi}{6}\)
\(45^\circ=\frac{\pi}{4}\)
\(60^\circ=\frac{\pi}{3}\)
Students should recognize the radian and degree form of each special angle immediately.
Using the Exact-Value Table
Find the required angle in the table and select the correct sine, cosine, or tangent value.
For reciprocal ratios, first use the corresponding sine, cosine, or tangent value.
For example:
\(\sec60^\circ=\frac{1}{\cos60^\circ}=2\)
\(\cot60^\circ=\frac{1}{\tan60^\circ}=\frac{1}{\sqrt3}\)
Rationalizing a Denominator
A radical should not normally remain in the denominator of a final exact answer.
For example:
\(\frac{1}{\sqrt3}\times\frac{\sqrt3}{\sqrt3}=\frac{\sqrt3}{3}\)
The value has not changed because the fraction used for multiplication is equal to one.
Evaluating Combined Expressions
Substitute the exact value of every ratio before simplifying.
For example:
\(\sin60^\circ\cos30^\circ+\cos60^\circ\sin30^\circ\)
becomes:
\(\frac{\sqrt3}{2}\cdot\frac{\sqrt3}{2}+\frac{1}{2}\cdot\frac{1}{2}=1\)
Products should be simplified before the terms are added or subtracted.
Expressions Containing 45 Degrees
Since:
\(\sin45^\circ=\cos45^\circ=\frac{1}{\sqrt2}\)
expressions containing both ratios can often be simplified by collecting like terms.
For example:
\(3\cos45^\circ+4\sin45^\circ=\frac{7}{\sqrt2}=\frac{7\sqrt2}{2}\)
Solutions of Exercise 6.5 Unit 6 Class 9 Math Notes
Exercise 6.5 applies trigonometric ratios and Pythagoras’ theorem to right-angled triangles. It includes special triangles, unknown sides and angles, field diagonals, canal width, compound figures, ladders, and algebraic length problems.
Solving a Right-Angled Triangle
To solve a triangle means to find all its unknown sides and angles.
First identify the hypotenuse and label the opposite and adjacent sides with respect to the known angle.
Select sine, cosine, or tangent according to the known and required sides.
Finding a Missing Side
Use:
\(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\)
when the opposite side and hypotenuse are involved.
Use:
\(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)
when the adjacent side and hypotenuse are involved.
Use:
\(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\)
when the two perpendicular sides are involved.
Finding a Missing Angle
Use an inverse trigonometric function after forming the correct ratio.
\(\theta=\sin^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)\)
\(\theta=\cos^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)\)
\(\theta=\tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right)\)
The remaining acute angle is found from:
\(A+C=90^\circ\)
The 30–60–90 Triangle
The sides opposite 30 degrees, 60 degrees, and 90 degrees are in the ratio:
\(1:\sqrt3:2\)
The shortest side is opposite 30 degrees, and the hypotenuse is twice the shortest side.
The 45–45–90 Triangle
The two perpendicular sides are equal.
The side ratio is:
\(1:1:\sqrt2\)
Therefore, the hypotenuse equals one leg multiplied by \(\sqrt2\).
Pythagoras’ Theorem
For a right-angled triangle:
\(a^2+b^2=c^2\)
Here, \(c\) is the hypotenuse.
Pythagoras’ theorem is used when two side lengths are known and the third side is required.
Diagonal of a Square or Rectangle
A diagonal divides a square or rectangle into two right-angled triangles.
For a square of side \(a\):
\(d=\sqrt{a^2+a^2}=a\sqrt2\)
For a rectangle:
\(d^2=l^2+w^2\)
Finding the Width of a Canal
A line drawn directly across a canal forms a right angle with the bank.
Pythagoras’ theorem is used to find the width when a diagonal distance and a distance along the bank are given.
An angle can then be found using an inverse trigonometric ratio and converted into radians where required.
Compound Right-Triangle Figures
Some diagrams contain two connected right triangles.
Solve the first triangle to find a shared side.
Use that side in the second triangle to find the required length.
Do not treat the complete figure as one triangle unless its sides form a single right triangle.
Rejecting a Negative Length
An algebraic equation may produce a positive and a negative root.
A negative root is rejected when the variable represents a physical length.
The reason for rejecting it should be stated in the solution.
Solutions of Exercise 6.6 Unit 6 Class 9 Math Notes
Exercise 6.6 covers Heights and Distances. Students draw right-triangle models for flag posts, trees, ladders, lighthouses, poles, cliffs, rivers, and kites and use angles of elevation or depression to find unknown measurements.
Angle of Elevation
An angle of elevation is measured upward from a horizontal line to an object above the observer.
The horizontal distance is usually the adjacent side, while the vertical height is the opposite side.
Angle of Depression
An angle of depression is measured downward from a horizontal line at the observer.
The horizontal lines at the observer and the lower object are parallel.
Therefore, the angle of depression equals the corresponding angle of elevation.
Drawing the Diagram
Represent vertical objects by vertical line segments and level ground by horizontal line segments.
Join the observation point to the top of the object to form the line of sight.
Mark the right angle and place the given angle at the correct observation point.
Selecting the Correct Ratio
Tangent is used most often in height-and-distance questions because it relates vertical height to horizontal distance.
\(\tan\theta=\frac{\text{height}}{\text{horizontal distance}}\)
Use sine when the height and line of sight are involved.
\(\sin\theta=\frac{\text{height}}{\text{line of sight}}\)
Use cosine when the horizontal distance and line of sight are involved.
\(\cos\theta=\frac{\text{horizontal distance}}{\text{line of sight}}\)
Height of a Vertical Object
For a flag post or building observed from level ground:
\(h=d\tan\theta\)
Here, \(d\) is the horizontal distance from the base and \(\theta\) is the angle of elevation.
Finding a Horizontal Distance
When the height and angle are known:
\(d=\frac{h}{\tan\theta}\)
This method is used in questions involving a ship viewed from a lighthouse and a river viewed from a hill.
Ladder and Kite Problems
A ladder or stretched kite string forms the hypotenuse of a right triangle.
If the vertical height is required, use sine.
\(h=L\sin\theta\)
If the horizontal distance is required, use cosine.
\(d=L\cos\theta\)
Two Observation Points
Some questions give two angles of elevation measured from points on the same straight line.
Assign a variable to the nearer distance.
The farther distance is the nearer distance plus the distance walked between the two observation points.
Form one tangent equation from each observation point and solve them simultaneously.
Width of a River
Find the horizontal distance from the hill or tower to the nearer shore.
Then find the distance to the farther shore.
The width of the river is the difference between these distances.
\(\text{River width}=d_{\text{far}}-d_{\text{near}}\)
Using a Calculator
Non-special angles such as 28 degrees, 50 degrees, or 70 degrees require a calculator.
Keep extra decimal places during the calculation and round only the final answer.
Make sure the calculator is in degree mode when the angle is given in degrees.
Solutions of Review Exercise 6 Unit 6 Class 9 Math Notes
Review Exercise 6 revises the complete Trigonometry unit. It includes multiple-choice questions, degree-radian conversions, identity proofs, remaining trigonometric ratios, and practical height-and-distance problems.
Multiple-Choice Questions
The multiple-choice section checks inverse trigonometric values, exact values, Pythagorean identities, co-function identities, degree-radian conversions, and simple height questions.
Students should identify the required formula before checking the available options.
Converting Degrees and Radians
The review includes angles written in whole degrees, decimal degrees, and degrees with minutes.
Use:
\(\text{Radians}=\text{Degrees}\times\frac{\pi}{180}\)
and:
\(\text{Degrees}=\text{Radians}\times\frac{180}{\pi}\)
Convert a decimal part of a degree into minutes by multiplying it by 60.
Proving Identities
The review asks students to prove identities using conjugates, reciprocal ratios, quotient ratios, and Pythagorean identities.
Begin with one side and simplify it until the other side appears.
Every cancellation should be performed on factors, not separate terms.
Finding Remaining Trigonometric Ratios
The review gives one ratio and asks for the other five.
Assign side lengths from the given fraction, use Pythagoras’ theorem, and then write the required ratios.
The quadrant information determines whether the answers are positive or negative.
Distance from a Building
When a building height and angle of elevation are known, use:
\(d=\frac{h}{\tan\theta}\)
The review applies this method to a 30-metre building observed at an angle of 28 degrees.
Height Reached by a Ladder
When the ladder length and angle with the ground are known, the ladder is the hypotenuse.
Use:
\(h=L\sin\theta\)
A clear diagram helps prevent the opposite and adjacent sides from being interchanged.
Important Rules of Unit 6 Class 9 Math
Degree-Minute-Second Relations
\(1^\circ=60′,\qquad1’=60”\)
Decimal Degrees
\(\text{Decimal degrees}=D+\frac{M}{60}+\frac{S}{3600}\)
Degrees to Radians
\(\text{Radians}=\text{Degrees}\times\frac{\pi}{180}\)
Radians to Degrees
\(\text{Degrees}=\text{Radians}\times\frac{180}{\pi}\)
Arc Length
\(s=r\theta\)
Sector Area
\(A=\frac{1}{2}r^2\theta\)
Sine Ratio
\(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\)
Cosine Ratio
\(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\)
Tangent Ratio
\(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\)
Pythagorean Identities
\(\sin^2\theta+\cos^2\theta=1\)
\(1+\tan^2\theta=\sec^2\theta\)
\(1+\cot^2\theta=\cosec^2\theta\)
Pythagoras’ Theorem
\(a^2+b^2=c^2\)
Common Mistakes in Unit 6 Class 9 Math Notes
Students sometimes confuse clockwise negative angles with anticlockwise positive angles.
A large or negative angle should be reduced to a suitable co-terminal angle before its quadrant is identified.
Minutes and seconds should not be treated as decimal parts of a degree without conversion.
The formulas \(s=r\theta\) and \(A=\frac{1}{2}r^2\theta\) require the angle to be in radians.
Students may interchange the opposite and adjacent sides because they do not label the triangle with respect to the given angle.
The reciprocal of sine is cosecant, not secant.
The signs of trigonometric ratios are sometimes ignored when the angle lies outside Quadrant I.
Exact values should not be replaced by decimal approximations unless the question asks for an approximate answer.
A radical denominator should be rationalized when an exact simplified answer is required.
While proving an identity, students sometimes work on both sides at the same time and assume the required result.
Terms connected by addition or subtraction cannot be cancelled directly.
The hypotenuse must always be the side opposite the right angle and the longest side of the triangle.
A calculator may be left in radian mode when the question gives an angle in degrees.
In height-and-distance questions, the angle of depression should be transferred correctly using parallel horizontal lines.
Intermediate values should not be rounded too early because this can change the final answer.
Exam Preparation Tips for Unit 6 Class 9 Math Notes
Memorize the degree-radian conversion formulas.
Learn the exact sine, cosine, and tangent values of 30 degrees, 45 degrees, and 60 degrees.
Memorize the reciprocal, quotient, and Pythagorean identities.
Draw a labelled right triangle before using a trigonometric ratio.
Practise identifying quadrants and signs of ratios.
Use Pythagoras’ theorem confidently to find missing sides.
For an identity, start from the more complicated side and simplify one step at a time.
Practise rationalizing denominators containing square roots.
Draw clean diagrams for heights-and-distances questions.
Check whether the calculator should be in degree or radian mode.
Keep exact values throughout a solution and round only when required.
Attempt Review Exercise 6 without viewing the solution PDF first.
Why Unit 6 Class 9 Math Solutions Are Important
Unit 6 Class 9 Math Solutions are important because trigonometry connects angles with lengths.
It is used in geometry, surveying, construction, engineering, navigation, astronomy, physics, computer graphics, and many other fields.
The unit also strengthens algebraic manipulation through identity proofs and improves problem-solving skills through practical right-triangle applications.
A strong understanding of Unit 6 prepares students for coordinate geometry, advanced trigonometry, calculus, and higher mathematics.
FAQs About Unit 6 Class 9 Math Solutions
What is the topic of Unit 6 Class 9 Math?
The topic of Unit 6 Class 9 Math is Trigonometry.
How many exercises are included in Unit 6?
Unit 6 includes Exercise 6.1, Exercise 6.2, Exercise 6.3, Exercise 6.4, Exercise 6.5, Exercise 6.6, and Review Exercise 6.
What is covered in Exercise 6.1?
Exercise 6.1 covers quadrants, co-terminal angles, degrees, minutes, seconds, radians, arc length, sector area, percentages of circles, and cone formation.
What is covered in Exercise 6.2?
Exercise 6.2 covers trigonometric ratios, signs in quadrants, exact values, right triangles, remaining ratios, and basic identities.
What is covered in Exercise 6.3?
Exercise 6.3 covers remaining trigonometric ratios and proofs of trigonometric identities.
What is covered in Exercise 6.4?
Exercise 6.4 covers exact trigonometric values of special angles and the evaluation of expressions without a calculator.
What is covered in Exercise 6.5?
Exercise 6.5 covers right-angled triangles, special triangles, Pythagoras’ theorem, unknown sides and angles, and practical geometry problems.
What is covered in Exercise 6.6?
Exercise 6.6 covers angles of elevation and depression and practical height-and-distance problems.
What are the three basic trigonometric ratios?
The three basic ratios are \(\sin\theta\), \(\cos\theta\), and \(\tan\theta\).
When can the formula s equals r theta be used?
The formula is used for arc length when the central angle is measured in radians.
What is the first step in a height-and-distance problem?
The first step is to draw and label a right-triangle diagram from the information given.
How should a trigonometric identity be proved?
Begin with one side, normally the more complicated side, and simplify it using valid identities and algebraic steps until the other side is obtained.
Are these Unit 6 Class 9 Math Solutions available in PDF format?
Yes. The solutions of Exercises 6.1 to 6.6 and Review Exercise 6 are available in PDF format on this page.
Are these solutions useful for exam preparation?
Yes. Students can use these solutions for homework, revision, class tests, annual examinations, and board exam preparation.
Disclaimer
These Unit 6 Class 9 Math Solutions are provided for educational help. Students should use them to understand the method, check their work, and study alongside the official textbook and their teacher’s instructions.
Final Words
Unit 6 Class 9 Math Solutions help students understand Trigonometry in a clear and organized way.
Exercise 6.1 develops angle and sector concepts, Exercises 6.2 to 6.4 build trigonometric ratios and identities, and Exercises 6.5 and 6.6 apply those ideas to right triangles and real-life measurements.
Study the exercises in order, practise the exact values and identities regularly, draw every diagram carefully, and use Review Exercise 6 to test your understanding of the complete unit.