Chapter 6: Q. 21 (page 435)

In Problems 16–23, find the exact value of each of the remaining trigonometric functions.
$\tan θ = \frac{1}{3}, 180^{°} < θ < 270^{°}$

Short Answer

Expert verified

The exact values of the remaining trigonometric functions of $\tan θ = \frac{1}{3}, 180^{°} < θ < 270^{°}$ are,

$\sin θ = - \frac{\sqrt{10}}{10} \cos θ = - \frac{3 \sqrt{10}}{10} c s c θ = - \sqrt{10} s e c θ = - \frac{\sqrt{10}}{3} c o t θ = 3$

Step by step solution

Step 1 Given expression is,

$\tan θ = \frac{1}{3}$ and $180^{°} < θ < 270^{°}$ .

So, $θ$ lies in the third quadrant and where $\tan θ$ is positive.

Let $α$ be the reference angle for $θ$ then, $\tan θ = \frac{b}{a} = \tan α$ .

So, $\tan α = \frac{1}{3} = \frac{o p p o s i t e}{a d j a c e n t}$ .

Use the values $b = 1, a = 3$ to construct a triangle.

From the figure, we obtain $r = \sqrt{10}$ .

Step 2 Now calculate the remaining trigonometric functions.

The exact values of the remaining trigonometric functions of the reference angle $α$ are,

$\sin α = \frac{b}{r} = \frac{1}{\sqrt{10}} \cos α = \frac{a}{r} = \frac{3}{\sqrt{10}} c s c α = \frac{r}{b} = \frac{\sqrt{10}}{1} = \sqrt{10} s e c α = \frac{\sqrt{10}}{3} c o t α = \frac{3}{1} = 3$

Step 3 The exact values of the trigonometric functions of the reference angle θ are,

$\sin θ = - \frac{\sqrt{10}}{10} \cos θ = - \frac{3 \sqrt{10}}{10} c s c θ = - \sqrt{10} s e c θ = - \frac{\sqrt{10}}{3} c o t θ = 3$

These are the exact values of the remaining trigonometric functions.

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In Problems 16–23, find the exact value of each of the remaining trigonometric functions.
$\tan θ = \frac{1}{3}, 180^{°} < θ < 270^{°}$

Short Answer

Step by step solution

Step 1 Given expression is,

Step 2 Now calculate the remaining trigonometric functions.

Step 3 The exact values of the trigonometric functions of the reference angle θ are,

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In Problems 16–23, find the exact value of each of the remaining trigonometric functions.tanθ=13,180°<θ<270°

Short Answer

Step by step solution

Step 1 Given expression is,

Step 2 Now calculate the remaining trigonometric functions.

Step 3 The exact values of the trigonometric functions of the reference angle θ are,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

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In Problems 16–23, find the exact value of each of the remaining trigonometric functions.
$\tan θ = \frac{1}{3}, 180^{°} < θ < 270^{°}$