Trigonometry Questions

Introduction to Trigonometry Questions

Trigonometry is a branch of mathematics that deals with the study of triangles, especially the right-angle triangle. The word "trigonometry" comes from the Greek words trigonon (triangle) and metron (to measure), which means measuring the triangle. In simple words, trigonometry helps us to understand the relationship between angles and sides of a triangle.

In trigonometry, we mainly use three important ratios called SINE (SIN), COSINE (COS), and Tangent (TAN). These ratios are used to find the unknown angles or sides of a right-angle triangle. The angle is usually represented with the symbol θ (Theta). Trigonometry is very useful in real life, for example, to find the height of a building, the distance of a ship from a lighthouse, or to find the angle of a ladder against a wall.

Common examples of trigonometry questions include finding a tree's height when the shadow is known, or finding the height angle when you look at the top of the tower. In this article, we will explore various types of trigonometry questions, their corresponding formulas, and solved examples, enabling students to understand the topic step by step.

 

Table of Contents

• What is Trigonometry?
• Trigonometry Questions and Answers
• Practice Questions
• Conclusion
• FAQs on Trigonometry Questions

 

What is Trigonometry?

Trigonometry is part of mathematics, studying the connection between the sides and the angles of a triangle. The word "trigonometry" comes from the Greek words "tri", meaning  three "gon" meaning sides, and "metron", meaning "measure". So 'trigonometry' means measuring a triangle.

In trigonometry, we work mostly with the right-angle triangle (a triangle that has an angle of 90 °). To solve questions, we use certain special ratios called trigonometric ratios. These ratios compare the length of the sides of a triangle with respect to an angle.

The most important trigonometric ratios are:

• Sine  (sin A) = Side opposite to ∠A ÷  Hypotenuse

• Cosine (cos A) = Side next to ∠A ÷  Hypotenuse

• Tangent (tan A) = Side opposite to ∠A ÷  Side next to ∠A

• Cosecant (cosec A) = Hypotenuse ÷  Side opposite to ∠A

• Secant (sec A) = Hypotenuse ÷  Side next to ∠A

• Cotangent (cot A) = Side next to ∠A ÷  Side opposite to ∠A

Also remember:

• tan A = sin A ÷ cos A

• cot A = cos A ÷ sin A

 

Trigonometry Questions and Answers

Question 1: Find the trigonometric ratios of a right triangle

In right ΔABC, ∠B = 90°. AB = 9 cm, BC = 12 cm, AC = 15 cm. Find tan A and cot C, and then calculate tan A − cot C.

Solution:

• At A: opposite = BC = 12, adjacent = AB = 9 ⇒ tan A = 12/9 = 4/3

• At C: opposite = AB = 9, adjacent = BC = 12 ⇒ cot C = adjacent/opposite = 12/9 = 4/3

• So, tan A − cot C = 4/3 − 4/3 = 0. 

Question 2: Find a missing side and a ratio

Right ΔPQR with ∠Q = 90°. PQ = 8 cm, QR = 6 cm. Find PR and sin P.

Solution:

• By Pythagoras: PR = √ ( PQ² + QR² ) = √ (8² + 6² ) = √ (64 + 36 )= √ 100 = 10 cm 
  

• At P: opposite = QR = 6, hypotenuse = PR = 10 ⇒ sin P = 6/10 = 3/5

Question 3: Height from shadow (angle of elevation)

A pole casts a shadow of 12 m when the Sun’s angle of elevation is 30°. Find the height of the pole.

Solution:

• tan 30° = (height / shadow) ⇒ 1/√ 3  = (h/12) ⇒ h = 12/√ 3 = 4√ 3 ≈ 3.93 m.

Question 4: Ladder against a wall

A 10 m ladder makes a 60° angle with the ground. Find the height it reaches on the wall and the distance of the foot from the wall.

Solution: 

• Height = 10 sin 60° = 10 . (√ 3/ 2) = 5√ 3 ≈ 8.66 m.

• Distance from wall = 10 cos 60° = 10. (1/2) = 5 m.

Question 5: Find the angle from a given ratio.

If sin θ = (1/2)​ and ,0° ≤ θ ≤ 90° find θ.

Solution: 30° = (1/2) ⇒ θ = 30°.

Question 6: Use complementary angles.

If cos x= (4/5)  for an acute angle x, find sin(90° - x)

Solution: sin (90° - x) = cos x = (4/5).

Question 7: Evaluate a simple identity value

Find the value of sin²  60° + cos² 60°.

Solution: sin  60° = (√ 3/2), cos 60° = (1/2),

(√ 3/2)$²\; +\; (1/2)²\; =\; (3/4)$  + (1/4) = 1.

Question 8: Convert from tan to sin and cos. 

If tan θ = (3/4) (acute θ), find sin⁡θ and cos⁡θ

Solution: Opposite: Adjacent: Hypotenuse = 3:4:5.

So, sin θ = 3/5 and cos θ = 4/5.

Question 9: Find an unknown angle in a right triangle.

In right ΔXYZ, ∠Y = 90°, sin X = (5/13, ), find ∠X and ∠Z.

Solution:

• sin X = 5/13 matches the 5 –12 –13 traingle ⇒ X ≈ sin^-1(5/13)
• Which is a standard ratio: X ≈ 22.62°
• Sum of angles in a triangle: X + Y + Z = 180°.
• So, Z = 180° - 90° - 22.62° = 67.38°. 

Question 10: Simpfly using 1 – cos² A = sin² A 

Show that (1 − cos² A ) / sin A = sin A (for A ≠ 0°,180°)

Solution: 

• 1 − cos² A = sin² A
• So, (sin² A / sin A) = sin A , Proved.
  
  

Practice Questions

1. Prove that: sin² 0 + cos² 0 = 1 

2. If sin A = (3/4) , find cos⁡A and tan⁡A.

3. Show that: 1 + tan² 0 = sec² 0

4. Simplify: ( 1 – cos² 0 / sin² 0 )

5. Evaluate: sin 30° + cos 60°

6. Prove that: cot ⁡θ ⋅ tan ⁡θ = 1

7. Simplify: sec² 0  – tan² 0

8. Prove that: sin 60° . cos 30° = ( √ 3 / 4)

9. Evaluate: sin² 45° + cos² 45° 

Conclusion

Trigonometry helps us to understand the link between angles and sides of a right triangle. By practicing trigonometry questions, students learn to use formulas like sine, cosine, and tangent to solve problems easily. Regular practice makes the topic simple, builds confidence, and shows how math is used in real life, like heights, distance, and angles.

 

FAQs on Trigonometry Questions

1. What are the 7 formulas of trigonometry?

Ans: The six trigonometric functions are sine, cosine, secant, cosecant, tangent, and cotangent. By using a right-angled triangle as a reference, the trigonometric functions and identities are derived: sin θ = Opposite Side/Hypotenuse. cos θ = Adjacent Side/Hypotenuse.

2. What is an example of a trigonometry question?

Ans: From a point on the ground 47 feet from the foot of a tree, the angle of elevation of the top of the tree is 35º. Find the height of the tree to the nearest foot.

3. Who is the father of trigonometry?

The father of trigonometry is generally considered to be Hipparchus. He lived in the 2nd century BC and is credited with creating the first trigonometric table, which was crucial for solving problems in astronomy.

4. Is trigonometry difficult?

Ans: Trigonometry can be challenging, but its difficulty varies depending on individual learning styles and prior mathematical knowledge. While some find it a daunting subject due to the introduction of new concepts like trigonometric functions and identities, others find it relatively straightforward with consistent practice and a good grasp of algebra.

5. What are the 6 basic trigonometries?

Ans: The six main trigonometric functions are sine, cosine, tangent, secant, cosecant, and cotangent. They are useful for finding heights and distances and have practical applications in many fields, including architecture, surveying, and engineering.