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Miscellaneous

Trigonometric Functions

**Example-1 :-** If sin x = 3/5, cos y = -12/13, where x and y both lie in second quadrant, find the value of sin (x + y).

We know that sin (x + y) = sin x cos y + cos x sin y ... (1) cos² x = 1 - sin² x = 1 - (3/5)² = 1 - 9/25 = 16/25 cos x = ± 4/5 Here, x lies on the second quadrant, so cos x is negative. cos x = -4/5 Now, sin² y = 1 - cos² y = 1 - (-12/13)² = 1 - 144/169 = 25/169 sin y = ± 5/13 Here, y lies on the second quadrant, so sin y is positive. sin y = 5/13 So, put the put the values of sin x, siny, cos x and cos y in (1) sin (x + y) = sin x cos y + cos x sin y = 3/5 x (-12/13) + (-4/5) x 5/13 = -36/65 + (-20/65) sin (x + y) = -56/65

**Example-2 :-** Prove that: cos 2x . cos x/2 - cos 3x . cos 9x/2 = sin 5x . sin 5x/2

**Example-3 :-** Find the value of tan π/8.

Let x = π/8, then 2x = π/4 Now, tan 2x = (2tan x)/(1 - tan² x) tan π/4 = (2tan π/8)/(1 - tan² π/8) Let y = tan π/8. Then, 1 = 2y/(1 - y²) 1 - y² = 2y y² + 2y - 1 = 0 By discrimination rule a = 1, b = 2, c = -1 d = b² - 4ac = 2² - 4 x 1 x (-1) = 4 + 4 d = 8 Now, y = (-b ± √d)/2a = (-2 ± √8)/(2 x 1) = (-2 ± 2√2)/2 = 2(-1 ± √2)/2 y = -1 ± √2 Therefore, π/8 lies on the first quadrant, y = tan π/8 is positive. Hence, tan π/8 = -1 + √2 = √2 - 1

**Example-4 :-** If tan x = 3/4, π < x < 3π/2, find the value of sin x/2, cos x/2 and tan x/2.

Since π < x < 3π/2 (3rd quadrant), cos x is negative Also, π/2 < x/2 < 3π/4 (2nd quadrant) Therefore, sin x/2 is positive and cos x/2 is negative. Now, sec² x = 1 + tan² x = 1 + (3/4)² = 1 + 9/16 = 25/16 sec x = 5/4 1/cos x = 5/4 cos x = 4/5 Here, x lies on the second quadrant, so cos x is negative. cos x = -4/5 Now, 2sin² x/2 = 1 - cos x = 1 - (-4/5) = 1 + 4/5 = 9/5 sin² x/2 = 9/10 sin x/2 = 3/√10 (π/2 < x/2 < 3π/4, sin x/2 is positive) Now, cos² x/2 = 1 - sin² x/2 = 1 - (3/√10)² = 1 - 9/10 cos² x/2 = 1/10 cos x/2 = 1/√10 cos x/2 = -1/√10 (π/2 < x/2 < 3π/4, cos x/2 is negative) Now, tan x/2 = (sin x/2)/(cos x/2) = (3/√10)/(-1/√10) = 3/√10 x (-√10/1) tan x/2 = -3

**Example-5 :-** Prove that: cos² x + cos² (x + π/3) + cos² (x - π/3) = 3/2

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