Evaluating Expressions Involving Inverse Trigonometric Functions. Evaluate each of the following expressions. sin-1(−√3/2) tan(tan-1 (−1/√3)) cos-1(cos(5π/4)) sin-1 (cos(2π/3))
Evaluating sin-1(−√3/2) is equivalent to finding the angle θ such that sinθ = −√3/2 and −π/2≤ θ ≤π/2. The angle θ = −π/3 satisfies these two conditions. Therefore, sin-1(−√3/2)=−π/3. First we use the fact that (tan-1 (−1/√3))=−π/6. Then tan(π/6)=−1/√3.. Therefore, tan(tan-1 (−1/√3))=−1/√3. To evaluate cos-1(cos(5π/4)) first use the fact that cos(5π/4)=−√2/2. Then we need to find the angle θ such that cos(θ)=−√2/2 and 0≤θ≤π. Since 3π/4 satisfies both these conditions, we have cos-1(cos(5π/4))=3π/4. Since cos(2π/3)=−1/2, we need to evaluate sin-1(−1/2). That is, we need to find the angle θ such that sin(θ)=−1/2 and −π/2≤θ≤π/2. Since −π/6 satisfies both these conditions, we can conclude that sin-1 (cos(2π/3))=−π/6.
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