Question

In Exercises 105-110, evaluate the expression without using a calculator. (Hint: Make a sketch of a right triangle, as illustrated in Example \(7 .)\) (a) \(\sin \left(\arcsin \frac{1}{2}\right)\) (b) \(\cos \left(\arcsin \frac{1}{2}\right)\)

Short answer

The value of \(\sin(\arcsin(1/2))\) is \(1/2\) and the value of \(\cos(\arcsin(1/2))\) is \(\sqrt{3}/2\).

Step-by-step solution

Evaluate \(\sin(\arcsin(1/2))\)

The arcsin of a number is the angle whose sin is that number. So, if we take the sin of the arcsin of a number, we should just get the number back. Therefore, \(\sin(\arcsin(1/2)) = 1/2.\)

Sketch a right triangle to find \(\cos(\arcsin(1/2))\)

Consider a right triangle where the side opposite the angle is 1 and the hypotenuse is 2. Then, by the Pythagorean theorem (\(a^2 + b^2 = c^2\)), the remaining side (adjacent to the angle and hence base of the triangle) is \(\sqrt{2^2 - 1^2} = \sqrt{3}\). The cosine of an angle in a right triangle is adjacent/hypotenuse = \(\sqrt{3}/2\). Therefore, the \(\cos(\arcsin(1/2)) = \sqrt{3}/2\)