Evaluate Using the Given Value cos(theta) if tan(theta)=-4/3

$\cos (θ)$ if $\tan (θ) = - \frac{4}{3}$

Step 1

Take the inverse tangent of both sides of the equation to extract $θ$ from inside the tangent.

$θ = \arctan (- \frac{4}{3})$

Step 2

Simplify the right side.

Evaluate $\arctan (- \frac{4}{3})$ .

$θ = - 0.92729521$

Step 3

The tangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the third quadrant.

$θ = - 0.92729521 - (3.14159265)$

Step 4

Simplify the expression to find the second solution.

Add $2 π$ to $- 0.92729521 - (3.14159265)$ .

$θ = - 0.92729521 - (3.14159265) + 2 π$

The resulting angle of $2.21429743$ is positive and coterminal with $- 0.92729521 - (3.14159265)$ .

$θ = 2.21429743$

Step 5

Find the period of $\tan (θ)$ .

The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Divide $π$ by $1$ .

$π$

Step 6

Add $π$ to every negative angle to get positive angles.

Add $π$ to $- 0.92729521$ to find the positive angle.

$- 0.92729521 + π$

Replace with decimal approximation.

$3.14159265 - 0.92729521$

Subtract $0.92729521$ from $3.14159265$ .

$2.21429743$

List the new angles.

$θ = 2.21429743$

Step 7

The period of the $\tan (θ)$ function is $π$ so values will repeat every $π$ radians in both directions.

$θ = 2.21429743 + π n, 2.21429743 + π n$ , for any integer $n$

Step 8

Consolidate $2.21429743 + π n$ and $2.21429743 + π n$ to $2.21429743 + π n$ .

$θ = 2.21429743 + π n$ , for any integer $n$

Step 9

Take the base solution.

$θ = 2.21429743$

Step 10

Replace the variable $θ$ with $2.21429743$ in the expression.

$\cos (2.21429743)$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\cos (2.21429743)$

Decimal Form:

$- 0.5\overline{9}$

Evaluate Using the Given Value cos(theta) if tan(theta)=-4/3

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