Question

Prove that Sin(90+x) is equal to Sin(180-x) and what is the value​

Answer 1

Answer:

From a rule of trigonometry,

sin(90-x) = cos(x)

You might have noticed from the trigonometric table that

sin(90–30) = cos30 = sin60 = 3^(1/2)/2

sin(90–60) = cos60 = sin30 = 1/2

sin(90–45) = cos(45) = sin45 1/(2^0.5).

Explanation(proof):see the image for construction

As you can see from above picture that In triangle ABC,

Measure of B = 90 Degrees

Therefore (A+C) = 180 - B = 90

Therefore, C= 90-A ……………(1)

Now, sinA =BC/AC (sin of any angle is equal to opposite side/hypotenuse)

sinC = AB/AC….(2)

cosA = AB/AC …(3) (cos of any angle is the ratio of adjacent side to hypotenuse)

From 2 and 3, sinC = cosA.

Now from 1, C = 90-A,

Therefore sinC = sin(90-A) = cosA.

Thus, sin(90-A) = cosA and vice versa, where A can be any angle.

Hope, this helps. This method an be applied to prove:

tan(90-A) = cotA, and vice versa;

cosec(90-A) = secA and vice versa.