4.3 Equivalent Trigonometric Expressions

Equivalent Trigonometric Expressions

Co-function Identities for Angles in the First Quadrant:

1) sin x = cos (π/2 – x) AND cos x = sin (π/2 – x)

2) tan x = cot (π/2 – x) AND cot x = tan (π/2 – x)

3) csc x = sec (π /2 – x) AND sec x = csc (π/2 – x)

Co-function Identities for Angles in the Second Quadrant:

1) sin (x + π/2) = cos x AND cos (x + π/2) = - sin x

2) tan (x + π/2) = - cot x AND cot (x + π/2) = - tan x

3) csc (x + π/2) = sec x AND sec (x + π/2) = - csc x

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sin 30° = cos 60°

sin 40° = cos 50°

sin 50° = cos 40°

sin 60° = cos 30°

Rule: the sum of the degrees should equal to 90°

sin(x) = cos (π/2 – x)

cos(x) = sin (π/2 – x)

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- sin 30° = cos 120°

- sin 40° = cos 130°

- sin 50° = cos 140°

- sin 60° = cos 150°

Rule: add 90° to – sin (x), subtract 90° to cos (x)

- sin(x) = cos (x + π/2)

cos(x) = - sin (x – π/2)

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Using Right Angle to Confirm:

sinx = a/b <

∠C = π/2 – x

sin ∠C = c/b

cos ∠C = a/b <

tan ∠C = c/a

Therefore,

sinx = cos ∠C

sinx = cos (π/2 – x)

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Example:

Given that sin (π/7) = 0.4338, find the value of cos (5π/14).

Finding the other angle (π/2 – x): 5π/14 = π/2 – x

Isolate x: x = π/2 – 5π/14

Common Denominator: x = 7π/14 – 5π/14

Simplify: x = 2π/14 = π/7

So,

cos (5π/14) = cos (π/2 – π/7) = sin (π/7) = 0.4338