Evaluate cos(sin^(-1)(1/11)), giving your answer as an exact value (no decimals)
To evaluate cos(sin^(-1)(1/11)) as an exact value (no decimals), we can use a right triangle. Let's go through the steps:
Step 1: Start with the inner function, sin^(-1)(1/11). This means we need to find an angle whose sine is equal to 1/11.
Step 2: Let's label our right triangle. Consider a right triangle with an opposite side of length 1 and a hypotenuse of length 11. The adjacent side is unknown and can be represented as 'x'.
Step 3: Using the Pythagorean theorem, we can solve for 'x'. According to Pythagorean theorem, a^2 + b^2 = c^2, where 'a' and 'b' are the legs of the triangle and 'c' is the hypotenuse. In this case, it will be x^2 + 1^2 = 11^2.
Simplifying the equation: x^2 + 1 = 121
x^2 = 120
x = √120
Step 4: Now that we have found the length of the adjacent side, we can evaluate cos(sin^(-1)(1/11)). Cosine is defined as the ratio of the adjacent side to the hypotenuse in a right triangle. In this case, it will be x/11.
Substituting the value of x: √120/11
Step 5: To simplify the expression, we can express √120 as a product of its prime factors: √(2^2 * 3 * 5) = 2√(3 * 5) = 2√15.
Final answer: cos(sin^(-1)(1/11)) = 2√15/11 (exact value).