Evaluate Using the Given Value cos(theta) if tan(theta)=3/2

$\cos (θ)$ if $\tan (θ) = \frac{3}{2}$

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

$θ = \arctan (\frac{3}{2})$

Simplify the right side.

Evaluate $\arctan (\frac{3}{2})$ .

$θ = 0.98279372$

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

$θ = (3.14159265) + 0.98279372$

Solve for $θ$ .

Remove parentheses.

$θ = 3.14159265 + 0.98279372$

Remove parentheses.

$θ = (3.14159265) + 0.98279372$

Add $3.14159265$ and $0.98279372$ .

$θ = 4.12438637$

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 $1$ .

$π$

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

$θ = 0.98279372 + π n, 4.12438637 + π n$ , for any integer $n$

Consolidate $0.98279372 + π n$ and $4.12438637 + π n$ to $0.98279372 + π n$ .

$θ = 0.98279372 + π n$ , for any integer $n$

Take the base solution.

$θ = 0.98279372$

Replace the variable $θ$ with $0.98279372$ in the expression.

$\cos (0.98279372)$

The result can be shown in multiple forms.

Exact Form:

$\cos (0.98279372)$

Decimal Form:

$0.55470019 \dots$

Evaluate Using the Given Value cos(theta) if tan(theta)=3/2

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