4.3 Equivalent Trigonometric Expressions

In this subtopic, we have learnt in Blended Learning on Monday. I find this subtopic quite confusing. Therefore, I tried to find some simple explanations for 4.3.



We have come up with the follwing identities:
1st Quadrant:
1. sinx = cos((π/2)-x)
2. cosx = sin((π/2)-x)

2nd Quadrant:
1. -sinx= cos(x+(π/2))
2. cosx= sin(x+(π/2))

key: subtract x from (π/2) and keep answer in radians
These identities connect the cosine and sine of an angle. This means that determining the x value in the cosine part of the equation will help you determine the x-value of sine. This method is used throughout the whole section of lesson 4.3.

Also, keep in mind this important aspect in order to get a better understanding of how these co-function identities in the 1st and 2nd quadrant are created.

cosx= sin(x+(π/2)): This means that doing a phase shift to the left (π/2) radians/units to the sine(x) function, creates the cos(x) function. Always think about transformations when dealing with all the other Co function identities as well. It happens for both the Co function identities for angles in the first quadrant and the Co function identities for angles in the second quadrant.


**I know the explanation is too wordy but try to understand though :)



In simpler manner, here's the trig cofunction identities :

For real number or angle measured in radians, theta, the following cofunction identities exist;

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

Hopefully this is helpful :)