CBSE Class 10 Mathematics Introduction to Trigonometry Notes Set 20

Measuring Angles

Angles are measured in degrees (a right angle is 90°, with 1° = 60′ and 1′ = 60″), in radians, and in grades (a right angle is 100 grades). Conversions: 180° = π radians = 200 grades, so 1 grade = 0.9° and radian measure = degrees × π/180.

Trigonometric Ratios

In a right triangle, for an acute angle θ: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent, and cosec θ, sec θ, cot θ are their reciprocals. The ratios depend only on the angle, not on the size of the triangle.
Standard values: sin 0°, 30°, 45°, 60°, 90° = 0, 1/2, 1/√2, √3/2, 1. cos takes the same values in reverse. tan 0°, 30°, 45°, 60° = 0, 1/√3, 1, √3, and tan 90° is not defined.

Identities and Complementary Angles

sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ, and tan θ = sin θ/cos θ.
Complementary angles: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, sec(90° − θ) = cosec θ. As a result tan θ · tan(90° − θ) = 1, which makes products like tan 5° tan 10° ... tan 85° equal to 1.

Heights and Distances

The angle of elevation is the angle above the horizontal line of sight; the angle of depression is the angle below it. Draw the figure, mark the horizontal and vertical lines, and use tan θ = height/distance in each right triangle. With two sightings, form two equations and eliminate the unknown distance.

Solved Examples

Example 1. Convert 144° into radians.
Answer:
144 × π/180 = 4π/5.
4π/5

Example 2. If sin A = 3/5, find tan A and sec A (A acute).
Answer:
The sides are 3, 4, 5, so cos A = 4/5.
tan A = 3/4 and sec A = 5/4

Example 3. Evaluate sin 30° cos 60° + cos 30° sin 60°.
Answer:
= (1/2)(1/2) + (√3/2)(√3/2) = 1/4 + 3/4.
1

Example 4. Prove that (sin θ + cosec θ)² + (cos θ + sec θ)² = 7 + tan²θ + cot²θ.
Answer:
Expand: sin²θ + 2 + cosec²θ + cos²θ + 2 + sec²θ = 1 + 4 + (1 + cot²θ) + (1 + tan²θ).
= 7 + tan²θ + cot²θ. Proved.

Example 5. Evaluate cos 20°/sin 70° + sin 35°/cos 55°.
Answer:
cos 20° = sin 70° and sin 35° = cos 55°.
1 + 1 = 2

Example 6. A ladder 10 m long leans against a wall making 60° with the ground. How high does it reach?
Answer:
Height = 10 sin 60°.
5√3 m

Example 7. From a point 50 m from the foot of a tower, the angle of elevation of its top is 30°. Find the height.
Answer:
Height = 50 tan 30° = 50/√3.
About 28.9 m

Example 8. The angle of elevation of the top of a tower is 60° from a point A and 45° from a point B, 10 m closer to the tower on the same line. Find the height.
Answer:
h/tan 45° − h/tan 60° = 10, so h(1 − 1/√3) = 10 and h = 10√3/(√3 − 1).
h = 5√3(√3 + 1) = 15 + 5√3 ≈ 23.66 m

Example 9. If tan θ = 3/4, find (sin θ + cos θ)/(sin θ − cos θ).
Answer:
Divide by cos θ: (tan θ + 1)/(tan θ − 1) = (7/4)/(−1/4).
−7

Example 10. If 3 sin θ = 2 cos²θ, find the acute angle θ.
Answer:
3 sin θ = 2 − 2 sin²θ gives 2 sin²θ + 3 sin θ − 2 = 0, so (2 sin θ − 1)(sin θ + 2) = 0.
sin θ = 1/2, so θ = 30°.