Important Trigonometry Questions

Trigonometry can be tricky, especially for the students of class 10 and 11 when trigonometric ratios and identities are involved. In this article, we have collected a list of questions on all the important trigonometry concepts and formulas. Solving these questions, ranging from easy to hard will help you better your problem-solving skills in trigonometry.

Trigonometry Questions with Solutions

Question 1: Find the sine (sin) of an angle A in a right triangle, where the opposite side is 4 units and the hypotenuse is 5 units.

Solution:

Sine (sin) = Opposite / Hypotenuse = 4 / 5.

Question 2: Calculate the cosine (cos) of an angle B in a right triangle, given that the adjacent side is 3 units and the hypotenuse is 7 units.

Solution:

Cosine (cos) = Adjacent / Hypotenuse = 3 / 7.

Question 3: Determine the tangent (tan) of an angle C in a right triangle, where the opposite side is 6 units and the adjacent side is 8 units.

Solution:

Tangent (tan) = Opposite / Adjacent = 6 / 8 = 3 / 4.

Question 4: Find the value of sin(x) when x = 30 degrees (π/6 radians).

Solution:

sin(30°) = sin(π/6) = 1/2.

Question 5: Calculate the cosine (cos) of an angle θ in a right triangle, where the adjacent side is 5 units and the hypotenuse is 10 units.

Solution:

Cosine (cos) = Adjacent / Hypotenuse = 5 / 10 = 1/2.

Question 6: Find the sine (sin) of an angle α in a right triangle, where the opposite side is 6 units and the hypotenuse is 10 units.

Solution:

Sine (sin) = Opposite / Hypotenuse = 6 / 10 = 3/5.

Question 7: Calculate the cosine (cos) of an angle β in a right triangle, given that the adjacent side is 8 units and the hypotenuse is 15 units.

Solution:

Cosine (cos) = Adjacent / Hypotenuse = 8 / 15.

Question 8: Determine the tangent (tan) of an angle γ in a right triangle, where the opposite side is 7 units and the adjacent side is 24 units.

Solution:

Tangent (tan) = Opposite / Adjacent = 7 / 24.

Question 9: Find the value of sin(x) when x = 45 degrees (π/4 radians).

Solution:

sin(45°) = sin(π/4) = √2/2.

Question 10: Calculate the cosine (cos) of an angle θ in a right triangle, where the adjacent side is 5 units and the hypotenuse is 13 units.

Solution:

Cosine (cos) = Adjacent / Hypotenuse = 5 / 13.

Question 11: In a right triangle, the opposite side is 8 units, and the adjacent side is 15 units. Calculate the values of sine (sin), cosine (cos), and tangent (tan) of the angle θ.

Solution:

Sine (sin) = Opposite / Hypotenuse = 8 / 17.
Cosine (cos) = Adjacent / Hypotenuse = 15 / 17.
Tangent (tan) = Opposite / Adjacent = 8 / 15.

Question 12: If sin(θ) = 0.6 and θ is in the first quadrant, find cos(θ).

Solution:

Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1.
sin(θ) = 0.6, so sin²(θ) = 0.6² = 0.36.
Thus, cos²(θ) = 1 – 0.36 = 0.64.
Since θ is in the first quadrant, cos(θ) is positive.
Therefore, cos(θ) = √0.64 = 0.8.

Question 13: Find the exact value of tan(45° + 30°).

Solution:

Use the tangent sum formula: tan(A + B) = (tan(A) + tan(B)) / (1 – tan(A)tan(B)).
Here, A = 45° and B = 30°.
tan(45°) = 1 and tan(30°) = √3/3.
Therefore, tan(45° + 30°) = (1 + √3/3) / (1 – 1×√3/3).
Simplify the expression: (1 + √3/3) / (1 – √3/3).
This equals (√3 + 1) / (1 – √3).

Question 14: If the angle of elevation from a point on the ground to the top of a tower is 30°, and the distance from the point to the base of the tower is 40 meters, find the height of the tower.

Solution:

The situation forms a right triangle with the tower as the height.
Use the tangent function: tan(θ) = opposite/adjacent.
Here, θ = 30°, adjacent side (distance to the tower) = 40 m.
So, tan(30°) = height / 40.
tan(30°) is √3/3.
Therefore, height = 40 × √3/3 ≈ 23.09 meters.

Question 15: Solve for x: 2 sin(x) = √2, for 0° ≤ x ≤ 360°.

Solution:

First, isolate sin(x): sin(x) = √2 / 2.
The solutions to sin(x) = √2 / 2 are x = 45° and x = 135°.
So, the values of x that satisfy 2 sin(x) = √2 in the given range are x = 45° and x = 135°.

Question 16: If the sides of a right triangle are in the ratio 3:4:5, and the shortest side is 9 cm, find the length of the hypotenuse.

Solution:

The sides of the triangle are in the ratio 3:4:5.
The shortest side (3 parts) is 9 cm.
Therefore, each part is 9 / 3 = 3 cm.
The hypotenuse (5 parts) is 5 × 3 cm = 15 cm.
Thus, the length of the hypotenuse is 15 cm.

Question 17: Prove that sin⁴x – cos⁴x = 1 – 2cos²x.

Solution:

Start with the left-hand side (LHS): sin⁴x – cos⁴x.
Use the identity sin²x + cos²x = 1.
Rewrite LHS: (sin²x)² – (cos²x)².
Apply the difference of squares: (sin²x + cos²x)(sin²x – cos²x).
Substitute 1 for sin²x + cos²x: 1(sin²x – cos²x).
Now, use sin²x = 1 – cos²x: 1 – 2cos²x.
Thus, sin⁴x – cos⁴x = 1 – 2cos²x, proving the identity.

Question 18: Find the exact value of sin(15°).

Solution:

Use the half-angle formula: sin(θ/2) = √((1 – cosθ)/2).
Here, θ = 30°, so sin(15°) = sin(30°/2).
cos(30°) = √3/2.
Apply the formula: sin(15°) = √((1 – √3/2)/2).
Simplify: sin(15°) = √((2 – √3)/4) = (√(2 – √3))/2.

Question 19: Solve for x: cos²x – sin²x = 1/2, for 0 ≤ x ≤ 2π.

Solution:

Use the Pythagorean identity: sin²x + cos²x = 1.
Substitute for sin²x: cos²x – (1 – cos²x) = 1/2.
Simplify: 2cos²x = 3/2.
cos²x = 3/4.
Find x: cos(x) = ±√(3/4) = ±√3/2.
Solve for x in the given range: x = π/6, 5π/6, 7π/6, 11π/6.
Explanation:
For cos(x) = √3/2, x = π/6 and 11π/6 in the range 0 ≤ x ≤ 2π.
For cos(x) = -√3/2, x = 5π/6 and 7π/6 in the range 0 ≤ x ≤ 2π.

Question 20: Prove that tan(θ)tan(60° – θ)tan(60° + θ) = tan³(θ).

Solution:

Start with the left-hand side (LHS): tan(θ)tan(60° – θ)tan(60° + θ).
Use the tan subtraction and addition formulas:
tan(60° – θ) = (tan(60°) – tan(θ))/(1 + tan(60°)tan(θ)).
tan(60° + θ) = (tan(60°) + tan(θ))/(1 – tan(60°)tan(θ)).
Multiply these expressions with tan(θ).
tan(60°) is √3.
Simplify the equation to show it equals tan³(θ).
After simplification, LHS becomes tan³(θ), proving the identity.

Conclusion

The questions listed above require a good understanding of trigonometry ratios and identities. The solutions to these questions would provide the students of class 10 and 11 with all the tricks and methods to use the trigonometry value table and formulas.