Trigonometry Examples

$\cos (2 a) = \frac{3}{5}$

Step 1

Use the definition of cosine to find the known sides of the unit circle right triangle. The quadrant determines the sign on each of the values.

$\cos (2 a) = \frac{adjacent}{hypotenuse}$

Step 2

Find the opposite side of the unit circle triangle. Since the adjacent side and hypotenuse are known, use the Pythagorean theorem to find the remaining side.

$Opposite = - \sqrt{{hypotenuse}^{2} - {adjacent}^{2}}$

Step 3

Replace the known values in the equation.

$Opposite = - \sqrt{{(5)}^{2} - {(3)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Negate $\sqrt{{(5)}^{2} - {(3)}^{2}}$ .

Opposite $= - \sqrt{{(5)}^{2} - {(3)}^{2}}$

Step 4.2

Raise $5$ to the power of $2$ .

Opposite $= - \sqrt{25 - {(3)}^{2}}$

Step 4.3

Raise $3$ to the power of $2$ .

Opposite $= - \sqrt{25 - 1 \cdot 9}$

Step 4.4

Multiply $- 1$ by $9$ .

Opposite $= - \sqrt{25 - 9}$

Step 4.5

Subtract $9$ from $25$ .

Opposite $= - \sqrt{16}$

Step 4.6

Rewrite $16$ as $4^{2}$ .

Opposite $= - \sqrt{4^{2}}$

Step 4.7

Pull terms out from under the radical, assuming positive real numbers.

Opposite $= - 1 \cdot 4$

Step 4.8

Multiply $- 1$ by $4$ .

Opposite $= - 4$

Step 5

Find the value of sine.

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Step 5.1

Use the definition of sine to find the value of $\sin (2 a)$ .

$\sin (2 a) = \frac{opp}{hyp}$

Step 5.2

Substitute in the known values.

$\sin (2 a) = \frac{- 4}{5}$

Step 5.3

Move the negative in front of the fraction.

$\sin (2 a) = - \frac{4}{5}$

Step 6

Find the value of tangent.

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Step 6.1

Use the definition of tangent to find the value of $\tan (2 a)$ .

$\tan (2 a) = \frac{opp}{adj}$

Step 6.2

Substitute in the known values.

$\tan (2 a) = \frac{- 4}{3}$

Step 6.3

Move the negative in front of the fraction.

$\tan (2 a) = - \frac{4}{3}$

Step 7

Find the value of cotangent.

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Step 7.1

Use the definition of cotangent to find the value of $\cot (2 a)$ .

$\cot (2 a) = \frac{adj}{opp}$

Step 7.2

Substitute in the known values.

$\cot (2 a) = \frac{3}{- 4}$

Step 7.3

Move the negative in front of the fraction.

$\cot (2 a) = - \frac{3}{4}$

Step 8

Find the value of secant.

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Step 8.1

Use the definition of secant to find the value of $\sec (2 a)$ .

$\sec (2 a) = \frac{hyp}{adj}$

Step 8.2

Substitute in the known values.

$\sec (2 a) = \frac{5}{3}$

Step 9

Find the value of cosecant.

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Step 9.1

Use the definition of cosecant to find the value of $\csc (2 a)$ .

$\csc (2 a) = \frac{hyp}{opp}$

Step 9.2

Substitute in the known values.

$\csc (2 a) = \frac{5}{- 4}$

Step 9.3

Move the negative in front of the fraction.

$\csc (2 a) = - \frac{5}{4}$

Step 10

This is the solution to each trig value.

$\sin (2 a) = - \frac{4}{5}$

$\cos (2 a) = \frac{3}{5}$

$\tan (2 a) = - \frac{4}{3}$

$\cot (2 a) = - \frac{3}{4}$

$\sec (2 a) = \frac{5}{3}$

$\csc (2 a) = - \frac{5}{4}$