trigonometry questions with answers

Trigonometry Questions

Trigonometry questions given here involve finding the missing sides of a triangle with the help of trigonometric ratios and proving trigonometry identities. We know that trigonometry is one of the most important chapters of Class 10 Maths. Hence, solving these questions will help you to improve your problem-solving skills.

What is Trigonometry?

The word ‘trigonometry’ is derived from the Greek words ‘tri’ (meaning three), ‘gon’ (meaning sides) and ‘metron’ (meaning measure). Trigonometry is the study of relationships between the sides and angles of a triangle.

The basic trigonometric ratios are defined as follows.

sine of ∠A = sin A = Side opposite to ∠A/ Hypotenuse

cosine of ∠A = cos A = Side adjacent to ∠A/ Hypotenuse

tangent of ∠A = tan A = (Side opposite to ∠A)/ (Side adjacent to ∠A)

cosecant of ∠A = cosec A = 1/sin A = Hypotenuse/ Side opposite to ∠A

secant of ∠A = sec A = 1/cos A = Hypotenuse/ Side adjacent to ∠A

cotangent of ∠A = cot A = 1/tan A = (Side adjacent to ∠A)/ (Side opposite to ∠A)

Also, tan A = sin A/cos A

cot A = cos A/sin A

Also, read: Trigonometry

Trigonometry Questions and Answers

1. From the given figure, find tan P – cot R.

From the given,

In the right triangle PQR, Q is right angle.

By Pythagoras theorem,

PR 2 = PQ 2 + QR 2

QR 2 = (13) 2 – (12) 2

= 169 – 144

tan P = QR/PQ = 5/12

cot R = QR/PQ = 5/12

So, tan P – cot R = (5/12) – (5/12) = 0

2. Prove that (sin 4 θ – cos 4 θ +1) cosec 2 θ = 2

L.H.S. = (sin 4 θ – cos 4 θ +1) cosec 2 θ

= [(sin 2 θ – cos 2 θ) (sin 2 θ + cos 2 θ) + 1] cosec 2 θ

Using the identity sin 2 A + cos 2 A = 1,

= (sin 2 θ – cos 2 θ + 1) cosec 2 θ

= [sin 2 θ – (1 – sin 2 θ) + 1] cosec 2 θ

= 2 sin 2 θ cosec 2 θ

= 2 sin 2 θ (1/sin 2 θ)

3. Prove that (√3 + 1) (3 – cot 30°) = tan 3 60° – 2 sin 60°.

LHS = (√3 + 1)(3 – cot 30°)

= (√3 + 1)(3 – √3)

= 3√3 – √3.√3 + 3 – √3

= 2√3 – 3 + 3

RHS = tan 3 60° – 2 sin 60°

= (√3) 3 – 2(√3/2)

= 3√3 – √3

Therefore, (√3 + 1) (3 – cot 30°) = tan 3 60° – 2 sin 60°.

Hence proved.

4. If tan(A + B) = √3 and tan(A – B) = 1/√3 ; 0° < A + B ≤ 90°; A > B, find A and B.

tan(A + B) = √3

tan(A + B) = tan 60°

A + B = 60°….(i)

tan(A – B) = 1/√3

tan(A – B) = tan 30°

A – B = 30°….(ii)

Adding (i) and (ii),

A + B + A – B = 60° + 30°

Substituting A = 45° in (i),

45° + B = 60°

B = 60° – 45° = 15°

Therefore, A = 45° and B = 15°.

5. If sin 3A = cos (A – 26°), where 3A is an acute angle, find the value of A.

sin 3A = cos(A – 26°); 3A is an acute angle

cos(90° – 3A) = cos(A – 26°) {since cos(90° – A) = sin A}

⇒ 90° – 3A = A – 26

⇒ 3A + A = 90° + 26°

⇒ 4A = 116°

⇒ A = 116°/4

6. If A, B and C are interior angles of a triangle ABC, show that sin (B + C/2) = cos A/2.

We know that, for a given triangle, the sum of all the interior angles of a triangle is equal to 180°

A + B + C = 180° ….(1)

B + C = 180° – A

Dividing both sides of this equation by 2, we get;

⇒ (B + C)/2 = (180° – A)/2

⇒ (B + C)/2 = 90° – A/2

Take sin on both sides,

sin (B + C)/2 = sin (90° – A/2)

⇒ sin (B + C)/2 = cos A/2 {since sin(90° – x) = cos x}

7. If tan θ + sec θ = l, prove that sec θ = (l 2 + 1)/2l.

tan θ + sec θ = l….(i)

We know that,

sec 2 θ – tan 2 θ = 1

(sec θ – tan θ)(sec θ + tan θ) = 1

(sec θ – tan θ) l = 1 {from (i)}

sec θ – tan θ = 1/l….(ii)

tan θ + sec θ + sec θ – tan θ = l + (1/l)

2 sec θ = (l 2 + 1)l

sec θ = (l 2 + 1)/2l

8. Prove that (cos A – sin A + 1)/ (cos A + sin A – 1) = cosec A + cot A, using the identity cosec 2 A = 1 + cot 2 A.

LHS = (cos A – sin A + 1)/ (cos A + sin A – 1)

Dividing the numerator and denominator by sin A, we get;

= (cot A – 1 + cosec A)/(cot A + 1 – cosec A)

Using the identity cosec 2 A = 1 + cot 2 A ⇒ cosec 2 A – cot 2 A = 1,

= [cot A – (cosec 2 A – cot 2 A) + cosec A]/ (cot A + 1 – cosec A)

= [(cosec A + cot A) – (cosec A – cot A)(cosec A + cot A)] / (cot A + 1 – cosec A)

= cosec A + cot A

9. Prove that: (cosec A – sin A)(sec A – cos A) = 1/(tan A + cot A)

[Hint: Simplify LHS and RHS separately]

LHS = (cosec A – sin A)(sec A – cos A)

= (cos 2 A/sin A) (sin 2 A/cos A)

= cos A sin A….(i)

RHS = 1/(tan A + cot A)

= (sin A cos A)/ (sin 2 A + cos 2 A)

= (sin A cos A)/1

= sin A cos A….(ii)

From (i) and (ii),

i.e. (cosec A – sin A)(sec A – cos A) = 1/(tan A + cot A)

10. If a sin θ + b cos θ = c, prove that a cosθ – b sinθ = √(a 2 + b 2 – c 2 ).

a sin θ + b cos θ = c

Squaring on both sides,

(a sin θ + b cos θ) 2 = c 2

a 2 sin 2 θ + b 2 cos 2 θ + 2ab sin θ cos θ = c 2

a 2 (1 – cos 2 θ) + b 2 (1 – sin 2 θ) + 2ab sin θ cos θ = c 2

a 2 – a 2 cos 2 θ + b 2 – b 2 sin 2 θ + 2ab sin θ cos θ = c 2

a 2 + b 2 – c 2 = a 2 cos 2 θ + b 2 sin 2 θ – 2ab sin θ cos θ

a 2 + b 2 – c 2 = (a cos θ – b sin θ ) 2

⇒ a cos θ – b sin θ = √(a 2 + b 2 – c 2 )

Video Lesson on Trigonometry

Practice Questions on Trigonometry

Solve the following trigonometry problems.

• Prove that (sin α + cos α) (tan α + cot α) = sec α + cosec α.
• If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B.
• If sin θ + cos θ = √3, prove that tan θ + cot θ = 1.
• Evaluate: 2 tan 2 45° + cos 2 30° – sin 2 60°
• Express cot 85° + cos 75° in terms of trigonometric ratios of angles between 0° and 45°.

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Trigonometry Quiz 13

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Trigonometry Quiz 1

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Frequently Asked Questions

Free Mathematics Tutorials

Free Trigonometry Questions and Problems

Free tutorials and problems on solving trigonometric equations, trigonometric identities and formulas can also be found. Java applets are used to explore, interactively, important topics in trigonometry such as graphs of the 6 trigonometric functions, inverse trigonometric functions, unit circle, angle and sine law.

Angles in Trigonometry

• Find the coterminal angle . An analytical tutorial on how to find coterminal angles.
• Find The Reference Angle. Analytical tutorial to find the reference angle to an angle.
• Table for the 6 trigonometric functions for special angles. A table of values of sin, cos, tan, csc, sec and cot for special angles 0, 30, 45, 60 and 90 degrees.
• Special Angles On Unit Circle. Special angles together with their sine and cosine displayed on a unit circle.
• Angle in Trigonometry . Understand the definition and properties of an angle in standard position
• Special Right Triangle used to find the six trigonomatric funtions of the special angles.

Trigonometric Functions

• Periods Of Trigonometric Functions . The periods of all 6 trigonometric functions are explored interactively using an applet.
• Properties of The Six Trigonometric Functions . The properties of the 6 trigonometric functions sin (x), cos (x), tan (x), cot (x), sec (x) and csc (x) are discussed. These include the graph, domain, range, asymptotes (if any), symmetry, x and y intercepts and maximum and minimum points.
• Sine Function . The sine function f(x) = a*sin(bx+c)+d is explored, interactively, using a large applet.
• Cosine Function . An applet helps you explore the general cosine function f(x) = a*cos(bx + c) + d.
• Tangent Function . The tangent function f(x) = a*tan(bx+c)+d and its properties such as graph, period, phase shift and asymptotes by changing the parameters a, b, c and d are explored interactively using an applet.
• Secant Function . The secant function f(x) = a*sec(bx+c)+d and its properties such as period, phase shift, asymptotes domain and range are explored using an interactive applet by changing the parameters a, b, c and d.
• Cosecant Function . The cosecant function f(x) = a * csc ( b x + c) + d and its period, phase shift, asymptotes, domain and range are explored using an applet.
• Cotangent Function . The cotangent function f(x) = a * cot ( b x + c) + d is explored along with its properties such as period, phase shift, asymptotes, domain and range.
• Graph of Sine, a*sin(bx+c), Function . Graphing and sketching sine functions of the form f (x) = a*sin (bx + c; step by step tutorial.
• Graphs of Basic Trigonometric Functions . The graphs and properties such as domain, range, vertical asymptotes of the 6 basic trigonometric functions: sin(x), cos(x), tan(x), cot(x), sec(x) and csc(x) are explored using an applet.
• Sum of Sine and Cosine Functions . An interactive tutorial to explore the sums involving sine and cosine functions such as f(x) = a*sin(bx)+ d*cos(bx).

Unit Circle in Trigonometry

• Unit Circle And The Trigonometric Functions sin(x), cos(x) and tan(x) . Using the unit circle, you will be able to explore and gain deep understanding of some of the properties, such as domain, range, asymptotes (if any) of the trigonometric functions.

Inverse Trigonometric Functions

• Inverse Trigonometric Functions . Inverse trigonometric functions are explored interactively using an applet.
• Graph, Domain and Range of Arccos function . The graph and the properties of the inverse trigonometric function arcsin are explored using an app.
• Graph, Domain and Range of Arctan function . The graph of the inverse trigonometric function arctan and its properties are explored using an app.
• Graph, Domain and Range of Arcsin function . The graph and the properties of the inverse trigonometric function arcsin are explored using an app.
• Solve Inverse Trigonometric Functions Questions . Questions on inverse trigonometric functions are solved and detailed solutions are presented. Also included are exercises with answers.
• Find Domain and Range of Arcsine Functions
• Find Domain and Range of Arccos Functions
• Trigonometry Angle Questions With Answers . Trigonometry questions related to angles in standard position, coterminal angles, complementary and supplementary angles, as well as conversion from degrees to radians and vice versa, are presented. The solutions and answers are provided.
• Rotation, Angular and Linear Speed - Questions with Answers . Questions related to angular and linear speeds of rotating objects are presented. The solutions and answers are also provided.
• Trigonometric Functions - Questions With Answers . Solve trigonometry questions related to trigonometric functions. The solutions and answers are provided.
• Simplify Trigonometric Expressions - Questions With Answers . Use trigonometric identities and formulas to simplify trigonometric expressions.
• Find Exact Values of Trigonometric Functions - Questions With Answers . Find exact values of trigonometric functions without using a calculator.

Solving Trigonometric Equations

• Trigonometric Equations . Tutorial with detailed explanations on how to solve trigonometric equations using different methods and strategies and the properties of trigonometric functions and identities.

Trigonometry Problems

• Solve Trigonometry Problems . A set of problems with detailed solutions are presented.
• Use Sine Functions to Model Problems . Tutorial on how to use sine functions to model problems. Given data and information about a certain situation, we model it in the form f(x) = A sin (b x + c) + D or f(x) = A cos (b x + c) + D .
• Solve Problems Using Trigonometric Ratios . A set of problems with detailed solutions are presented. These problems have been designed to reinforce the use of the trigonometric ratios and Pythagorean theorem.
• Tutorial on Sine Functions (1)- Problems . Tutorial on sine function problems. Examples with detailed solutions and explanations are included.
• Tutorial on Sine Functions (2)- Problems. This is a tutorial on the relationship between the amplitude, the vertical shift and the maximum and minimum of the sine function.

Trigonometric Identities and Their Applications

• Trigonometric Identities . A list of the basic trigonometric identities.
• Using Trigonometric Identities . How basic trigonometric identities are used? A tutorial with Several examples with detailed solutions is presented.
• Verify Trigonometric Identities . How to verify trigonometric identities? Several examples with detailed solutions are presented.

Trigonometric Formulas and Their Applications

• Sum, Difference and Product of Trigonometric Formulas Questions
• Sine and Cosine Sum of Angles From Euler's Formula

Polar Coordinates

• Convert Polar to Rectangular Coordinates and Vice Versa . Problems, with detailed solutions, where polar coordinates are converted into rectangular coordinates and vice versa are presented.

Problems and Self Tests

• Solve Trigonometric Equations . 10 problems, with their answers, on solving trigonometric equations are presented. These may be used as a self test on solving trigonometric equations.
• Test on Graphs Trigonometric Functions . A set of questions, with their answers, on identifying the graphs of trigonometric functions sin(x), cos(x), tan(x), sec(x), csc(x), cot(x) are presented. These may be used as a self test on the graphs of trigonometric functions.

Applications Of Trigonometry

• Sine Law - Ambiguous Case - applet . The ambiguous case of the sine law, in solving triangle problems, is explored interactively using an applet.
• Solve Right Triangle Problems with detailed solutions and explanations included.

Trigonometric Tables, Formulas and Worksheets

• Trigonometric tables . Trigonometric tables of all 6 trigonometric functions, with angles in degrees and radians. Copies of these tables can be downloaded.
• Trigonometric Identities and Formulas . Important definitions, identities and formulas used in trigonometry.
• Trigonometry Calculators . Several online trigonometry calculators and solvers in this site.
• Free trigonometry worksheets to download

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