Given 𝛼 = tan−1 𝑥, find the exact values of sin 𝛼 and cos 𝛼

Sure, let's solve this step by step.

Step 1: Understand the problem We are given that 𝛼 = tan^(-1) x. This means that 𝛼 is the angle whose tangent is x. We are asked to find the exact values of sin 𝛼 and cos 𝛼.

Step 2: Draw a right triangle To solve this problem, we can draw a right triangle where the angle 𝛼 is one of the non-right angles. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. So, we can label the opposite side as x (since tan 𝛼 = x) and the adjacent side as 1.

Step 3: Find the hypotenuse By the Pythagorean theorem, the hypotenuse is sqrt(x^2 + 1^2) = sqrt(x^2 + 1).

Step 4: Find sin 𝛼 and cos 𝛼 The sine of an angle in a right triangle is the ratio of the opposite side to the hypotenuse. So, sin 𝛼 = x/sqrt(x^2 + 1). The cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse. So, cos 𝛼 = 1/sqrt(x^2 + 1).

So, the exact values of sin 𝛼 and cos 𝛼 are x/sqrt(x^2 + 1) and 1/sqrt(x^2 + 1), respectively.