Question

Identify the exact value of the trigonometric functions sin x and cos x.

If x is an acute angle and tan x = 5,
The value of sin x = .
The value of cos x = .

Answer 1

The exact value of sin x is 5√(26)/26 and the exact value of cos x is √(26)/26 when tan x = 5.

Step-by-step explanation:

If x is an acute angle and tan x = 5, this means that the ratio of the opposite side to the adjacent side in a right triangle is 5. This is equivalent to saying that if the opposite side is represented by y, and the adjacent side is represented by x, then y/x = 5. To find the exact value of sin x and cos x, we can use the Pythagorean theorem, which states that in a right triangle with hypotenuse h, the sum of the squares of the other two sides (x and y) equals the square of the hypotenuse. Therefore, if we let x = 1 (the adjacent side), then the opposite side y = 5, and using the Pythagorean theorem, we have h = √(12 + 52) = √(1 + 25) = √(26). The exact values of sin x and cos x are then the ratios of the opposite side and adjacent side to the hypotenuse respectively.

sin x = y/h = 5/√(26) = 5√(26)/26

cos x = x/h = 1/√(26) = √(26)/26

Answer 2

For this case, we have the following expression:
so x = 5
Clearing x we have:
x = atan (5)
x = 78.69 degrees
We have then that the values for the sine and the cosine are:
sine (x) = sine (78.69) = 0.98
cosine (x) = cosine (78.69) = 0.19
Answer:
The value of sine x = 0.98
The value of cos x = 0.19