Chapter 6: Q. 55 (page 395)

Find the exact value of each of the remaining trigonometric functions of $θ .$
$tanθ = \frac{3}{4}, sinθ < 0$

Short Answer

Expert verified

The exact values of trigonometric functions are,

$secθ = - \frac{5}{4} cosθ = - \frac{4}{5} sinθ = - \frac{3}{5} cotθ = \frac{4}{3} cscθ = - \frac{5}{3}$

Step by step solution

Step 1. Given Information

Given trigonometric functions is $tanθ = \frac{3}{4}, sinθ < 0 .$ To find the exact values of trigonometric functions using substitution.

Step 2. Solve a equation

It implies that $cosθ < 0$ and angle is III quadrant. On using the equation $\tan^{2} θ + 1 = \sec^{2} θ \sec^{2} θ = \pm \sqrt{\tan^{2} θ + 1}$

Step 3. Substitution to find trigonometric function

On substituting the values,

$secθ = - \sqrt{1 + {(\frac{3}{4})}^{2}} = - \sqrt{1 + \frac{9}{16}} = - \sqrt{\frac{25}{16}} secθ = - \frac{5}{4} cosθ = \frac{1}{secθ} = \frac{1}{- \frac{5}{4}} cosθ = - \frac{4}{5}$

Step 4. Substitution for sin, cot and csc functions

On using the equation of $sinθ = \pm \sqrt{1 - \cos^{2} θ} sinθ = - \sqrt{1 - {(- \frac{4}{5})}^{2}} = - \sqrt{1 - \frac{16}{25}} sinθ = - \frac{3}{5} cotθ = \frac{1}{tanθ} = \frac{1}{\frac{3}{4}} cotθ = \frac{4}{3} cscθ = \frac{1}{sinθ} = \frac{1}{- \frac{3}{5}} cscθ = - \frac{5}{3}$

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Find the exact value of each of the remaining trigonometric functions of $θ .$
$tanθ = \frac{3}{4}, sinθ < 0$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solve a equation

Step 3. Substitution to find trigonometric function

Step 4. Substitution for sin, cot and csc functions

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Find the exact value of each of the remaining trigonometric functions of θ.tanθ=34,sinθ<0

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solve a equation

Step 3. Substitution to find trigonometric function

Step 4. Substitution for sin, cot and csc functions

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Probability and Statistics

Discrete Mathematics

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Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.

Find the exact value of each of the remaining trigonometric functions of $θ .$
$tanθ = \frac{3}{4}, sinθ < 0$