CBSE Class 10 Mathematics Introduction to Trigonometry Notes Set 18

Angles and Their Units

An angle is the amount a ray turns about its end point. Anticlockwise turning is positive and clockwise turning is negative.
Three systems are used: degrees (a right angle is 90°, with 1° = 60′ and 1′ = 60″), grades (a right angle is 100 grades) and radians (a straight angle is \( \pi \) radians).
Conversion: \( \frac{D}{90} = \frac{G}{100} = \frac{2R}{\pi} \). So \( 1^c = \frac{180^\circ}{\pi} \) and 1 grade = 0.9°.
Arc length: if an arc of length l subtends an angle \( \theta \) radians at the centre of a circle of radius r, then \( l = r\theta \). Equal arcs on two circles give \( r_1\theta_1 = r_2\theta_2 \).
Clock angles: the minute hand moves 6° per minute and the hour hand 0.5° per minute, so at h hours m minutes the angle between them is \( |30h - 5.5m| \) degrees (or 360° minus it).

Trigonometric Ratios and Standard Values

In a right triangle, for an acute angle \( \theta \): sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent, and cosec, sec, cot are their reciprocals.
Standard values: sin 0°, 30°, 45°, 60°, 90° = 0, 1/2, 1/√2, √3/2, 1. The cosine values run in the opposite order. tan 0°, 30°, 45°, 60° = 0, 1/√3, 1, √3.
Complementary angles: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, sec(90° − θ) = cosec θ. So tan 1° tan 2° ... tan 89° = 1, because each tan x pairs with tan(90° − x) to give 1.
Sum and difference: sin(A ± B) = sin A cos B ± cos A sin B and cos(A ± B) = cos A cos B ∓ sin A sin B. For example sin 15° = (√3 − 1)/(2√2) and cos 75° has the same value.

Identities and Solving Equations

Basic identities: \( \sin^2\theta + \cos^2\theta = 1 \), \( 1 + \tan^2\theta = \sec^2\theta \), \( 1 + \cot^2\theta = \operatorname{cosec}^2\theta \).
Useful consequences: (sec θ + tan θ)(sec θ − tan θ) = 1; sin⁴θ − cos⁴θ = sin²θ − cos²θ; sin⁴θ + cos⁴θ = 1 − 2 sin²θ cos²θ; sin⁶θ + cos⁶θ = 1 − 3 sin²θ cos²θ.
If a cos θ + b sin θ = c, then a sin θ − b cos θ = ±√(a² + b² − c²).
Proving an identity: work on the more complicated side, convert everything to sin and cos, combine fractions, and use the identities above. For equations on 0° ≤ θ ≤ 90°, factor or substitute t = sin θ, then reject values that lie outside the allowed range.
Bounds: tan²θ + cot²θ ≥ 2, and 1/2 ≤ sin⁴θ + cos⁴θ ≤ 1, with sin²θ cos²θ ≤ 1/4.