Question

Evaluate each expression without using a calculator.cos (tan⁻¹ (-2))

Accepted Answer

Recognize that the expression is $$ \cos(	an^{-1}(-2)) . $$Here, $$ 	an^{-1}(-2) $$ represents an angle $$ 	heta $$ whose tangent is $$ -2 . So$$, set $$ 	heta = 	an^{-1}(-2) $$, which means $$ 	an(	heta) = -2 . $$Visualize or draw a right triangle to represent the angle $$ 	heta . $$Since $$ 	an(	heta) = \frac{	ext{opposite}}{	ext{adjacent}} = -2 $$, assign the opposite side length as $$ -2 $$ and the adjacent side length as $$ 1  ($$the negative sign indicates direction, but for length use positive values and consider the quadrant later). Calculate the hypotenuse $$ h  of $$the triangle using the Pythagorean theorem: $$ h = \sqrt{(	ext{opposite})^2 + (	ext{adjacent})^2} = \sqrt{(-2)^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5} . $$Recall that $$ \cos(	heta) = \frac{	ext{adjacent}}{	ext{hypotenuse}} . $$Using the triangle, this becomes $$ \cos(	heta) = \frac{1}{\sqrt{5}} . $$Determine the sign of $$ \cos(	heta) $$ based on the quadrant of $$ 	heta . $$Since $$ 	an(	heta) = -2  is $$negative, $$ 	heta $$ lies either in the second or fourth quadrant. Cosine is positive in the fourth quadrant and negative in the second. Because the tangent is negative and the angle is the inverse tangent of a negative number, $$ 	heta  is in $$the fourth quadrant, so $$ \cos(	heta)  is $$positive. Therefore, $$ \cos(	an^{-1}(-2)) = \frac{1}{\sqrt{5}} .$$

Evaluate each expression without using a calculator.
cos (tan⁻¹ (-2))

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Concetti chiave

Inverse Tangent Function (tan⁻¹ or arctan)

Right Triangle Interpretation of Trigonometric Ratios

Relationship Between Sine, Cosine, and Tangent