Question

Find the exact value of each expression. Write the answer as a single fraction. Do not use a calculator. sin (3𝜋/2) tan (-15𝜋/4) - cos (-5𝜋/3)

Accepted Answer

First, simplify each trigonometric function by reducing the angles to their equivalent angles within the standard interval $$[0, 2\pi) or$$ $$[-\pi, \pi)$$ using the periodicity of sine, cosine, and tangent functions. For example, use the fact that $$\sin(	heta)$$ and $$\cos(	heta)$$ have period $$2\pi$$, and $$	an(	heta)$$ has period $$\pi. $$Calculate $$\sin 3\pi by $$recognizing that $$3\pi is $$equivalent to $$\pi$$ plus $$2\pi$$, so use the periodicity of sine: $$\sin(3\pi) = \sin(\pi). $$Simplify $$	an(-15\pi/4) by $$adding or subtracting multiples of $$\pi to $$bring the angle within the principal period of tangent, which is $$\pi. $$For example, add $$4\pi ($$which is $$16\pi/4) to$$ $$-15\pi/4 to $$get an equivalent positive angle. Simplify $$\cos(-5\pi/3) by $$using the even property of cosine, $$\cos(-	heta) = \cos(	heta)$$, and then reduce the angle $$5\pi/3 to an $$equivalent angle within $$[0, 2\pi) if $$necessary. After finding the exact values of $$\sin 3\pi$$, $$	an(-15\pi/4)$$, and $$\cos(-5\pi/3)$$, substitute them back into the expression $$\sin 3\pi 	an(-15\pi/4) - \cos(-5\pi/3)$$, and then write the result as a single fraction without using a calculator.

Find the exact value of each expression. Write the answer as a single fraction. Do not use a calculator. sin (3𝜋/2) tan (-15𝜋/4) - cos (-5𝜋/3)

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주요 개념

Unit Circle and Angle Measurement

Trigonometric Function Values for Special Angles

Trigonometric Identities and Simplification