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*)*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*)

- 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)

Using Right Angle to Confirm:

∠C = π/2 – *x*

sin ∠C = c/b

cos ∠C = a/b <

tan ∠C = c/a

Therefore,

sin*x* = cos ∠C

sin*x *= 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

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