Question

In Exercises 29–51, find the exact value of each expression. Do not use a calculator. tan [cos⁻¹ (− 4/5)]

Accepted Answer

Recognize that the expression is $$ 	an(\cos^{-1}(-\frac{4}{5})) . $$Here, $$ \cos^{-1}(-\frac{4}{5}) $$ represents an angle $$ 	heta $$ whose cosine is $$ -\frac{4}{5} . So$$, set $$ 	heta = \cos^{-1}(-\frac{4}{5}) $$, which means $$ \cos 	heta = -\frac{4}{5} . $$Recall the Pythagorean identity: $$ \sin^2 	heta + \cos^2 	heta = 1 . $$Use this to find $$ \sin 	heta  by $$substituting $$ \cos 	heta = -\frac{4}{5} $$:

$$ \sin^2 	heta = 1 - \cos^2 	heta = 1 - \left(-\frac{4}{5}ight)^2 = 1 - \frac{16}{25} = \frac{9}{25} $$

Then, $$ \sin 	heta = \pm \frac{3}{5} . $$Determine the correct sign of $$ \sin 	heta  by $$considering the range of $$ 	heta = \cos^{-1}(-\frac{4}{5}) . $$Since $$ \cos 	heta  is $$negative, $$ 	heta $$ lies in the second quadrant where sine is positive. Therefore, $$ \sin 	heta = \frac{3}{5} . $$Use the definition of tangent in terms of sine and cosine:

$$ 	an 	heta = \frac{\sin 	heta}{\cos 	heta} = \frac{\frac{3}{5}}{-\frac{4}{5}} $$ Simplify the fraction by dividing the numerators and denominators:

$$ 	an 	heta = \frac{3}{5} 	imes \frac{5}{-4} = -\frac{3}{4} $$

This gives the exact value of the original expression.

In Exercises 29–51, find the exact value of each expression. Do not use a calculator. tan [cos⁻¹ (− 4/5)]

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Inverse Cosine Function (cos⁻¹)

Right Triangle Trigonometry

Tangent Function and Sign Determination