Trigonometry Examples

sin (θ) = \frac{3}{4}

$\sin (θ) = \frac{3}{4}$

Step 1

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

sin (θ) = \frac{opposite}{hypotenuse}

$\sin (θ) = \frac{opposite}{hypotenuse}$

Step 2

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

Adjacent = \sqrt{{hypotenuse}^{2} - {opposite}^{2}}

$Adjacent = \sqrt{{hypotenuse}^{2} - {opposite}^{2}}$

Step 3

Replace the known values in the equation.

Adjacent = \sqrt{{(4)}^{2} - {(3)}^{2}}

$Adjacent = \sqrt{{(4)}^{2} - {(3)}^{2}}$

Step 4

Simplify inside the radical.

Tap for more steps...

Step 4.1

Raise

4

$4$ to the power of

2

$2$ .

Adjacent

= \sqrt{16 - {(3)}^{2}}

$= \sqrt{16 - {(3)}^{2}}$

Step 4.2

Raise

3

$3$ to the power of

2

$2$ .

Adjacent

= \sqrt{16 - 1 \cdot 9}

$= \sqrt{16 - 1 \cdot 9}$

Step 4.3

Multiply

- 1

$- 1$ by

9

$9$ .

Adjacent

= \sqrt{16 - 9}

$= \sqrt{16 - 9}$

Step 4.4

Subtract

9

$9$ from

16

$16$ .

Adjacent

= \sqrt{7}

$= \sqrt{7}$

Adjacent

= \sqrt{7}

$= \sqrt{7}$

Step 5

Find the value of cosine.

Tap for more steps...

Step 5.1

Use the definition of cosine to find the value of

cos (θ)

$\cos (θ)$ .

cos (θ) = \frac{adj}{hyp}

$\cos (θ) = \frac{adj}{hyp}$

Step 5.2

Substitute in the known values.

cos (θ) = \frac{\sqrt{7}}{4}

$\cos (θ) = \frac{\sqrt{7}}{4}$

cos (θ) = \frac{\sqrt{7}}{4}

$\cos (θ) = \frac{\sqrt{7}}{4}$

Step 6

Find the value of tangent.

Tap for more steps...

Step 6.1

Use the definition of tangent to find the value of

tan (θ)

$\tan (θ)$ .

tan (θ) = \frac{opp}{adj}

$\tan (θ) = \frac{opp}{adj}$

Step 6.2

Substitute in the known values.

tan (θ) = \frac{3}{\sqrt{7}}

$\tan (θ) = \frac{3}{\sqrt{7}}$

Step 6.3

Simplify the value of

tan (θ)

$\tan (θ)$ .

Tap for more steps...

Step 6.3.1

Multiply

\frac{3}{\sqrt{7}}

$\frac{3}{\sqrt{7}}$ by

\frac{\sqrt{7}}{\sqrt{7}}

$\frac{\sqrt{7}}{\sqrt{7}}$ .

tan (θ) = \frac{3}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}

$\tan (θ) = \frac{3}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 6.3.2

Combine and simplify the denominator.

Tap for more steps...

Step 6.3.2.1

Multiply

\frac{3}{\sqrt{7}}

$\frac{3}{\sqrt{7}}$ by

\frac{\sqrt{7}}{\sqrt{7}}

$\frac{\sqrt{7}}{\sqrt{7}}$ .

tan (θ) = \frac{3 \sqrt{7}}{\sqrt{7} \sqrt{7}}

$\tan (θ) = \frac{3 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 6.3.2.2

Raise

\sqrt{7}

$\sqrt{7}$ to the power of

1

$1$ .

tan (θ) = \frac{3 \sqrt{7}}{\sqrt{7} \sqrt{7}}

$\tan (θ) = \frac{3 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 6.3.2.3

Raise

\sqrt{7}

$\sqrt{7}$ to the power of

1

$1$ .

tan (θ) = \frac{3 \sqrt{7}}{\sqrt{7} \sqrt{7}}

$\tan (θ) = \frac{3 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 6.3.2.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

tan (θ) = \frac{3 \sqrt{7}}{{\sqrt{7}}^{1 + 1}}

$\tan (θ) = \frac{3 \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 6.3.2.5

Add

1

$1$ and

1

$1$ .

tan (θ) = \frac{3 \sqrt{7}}{{\sqrt{7}}^{2}}

$\tan (θ) = \frac{3 \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 6.3.2.6

Rewrite

{\sqrt{7}}^{2}

${\sqrt{7}}^{2}$ as

7

$7$ .

Tap for more steps...

Step 6.3.2.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{7}

$\sqrt{7}$ as

7^{\frac{1}{2}}

$7^{\frac{1}{2}}$ .

tan (θ) = \frac{3 \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}

$\tan (θ) = \frac{3 \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 6.3.2.6.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

tan (θ) = \frac{3 \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}

$\tan (θ) = \frac{3 \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 6.3.2.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

tan (θ) = \frac{3 \sqrt{7}}{7^{\frac{2}{2}}}

$\tan (θ) = \frac{3 \sqrt{7}}{7^{\frac{2}{2}}}$

Step 6.3.2.6.4

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 6.3.2.6.4.1

Cancel the common factor.

$\tan (θ) = \frac{3 \sqrt{7}}{7^{\frac{2}{2}}}$

Step 6.3.2.6.4.2

Rewrite the expression.

$\tan (θ) = \frac{3 \sqrt{7}}{7}$

Step 6.3.2.6.5

Evaluate the exponent.

$\tan (θ) = \frac{3 \sqrt{7}}{7}$

Step 7

Find the value of cotangent.

Tap for more steps...

Step 7.1

Use the definition of cotangent to find the value of

$\cot (θ)$ .

$\cot (θ) = \frac{adj}{opp}$

Step 7.2

Substitute in the known values.

$\cot (θ) = \frac{\sqrt{7}}{3}$

Step 8

Find the value of secant.

Tap for more steps...

Step 8.1

Use the definition of secant to find the value of

$\sec (θ)$ .

$\sec (θ) = \frac{hyp}{adj}$

Step 8.2

Substitute in the known values.

$\sec (θ) = \frac{4}{\sqrt{7}}$

Step 8.3

Simplify the value of

$\sec (θ)$ .

Tap for more steps...

Step 8.3.1

Multiply

$\frac{4}{\sqrt{7}}$ by

$\frac{\sqrt{7}}{\sqrt{7}}$ .

$\sec (θ) = \frac{4}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 8.3.2

Combine and simplify the denominator.

Tap for more steps...

Step 8.3.2.1

Multiply

$\frac{4}{\sqrt{7}}$ by

$\frac{\sqrt{7}}{\sqrt{7}}$ .

$\sec (θ) = \frac{4 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 8.3.2.2

Raise

$\sqrt{7}$ to the power of

$1$ .

$\sec (θ) = \frac{4 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 8.3.2.3

Raise

$\sqrt{7}$ to the power of

$1$ .

$\sec (θ) = \frac{4 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 8.3.2.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sec (θ) = \frac{4 \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 8.3.2.5

Add

$1$ and

$1$ .

$\sec (θ) = \frac{4 \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 8.3.2.6

Rewrite

${\sqrt{7}}^{2}$ as

$7$ .

Tap for more steps...

Step 8.3.2.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{7}$ as

$7^{\frac{1}{2}}$ .

$\sec (θ) = \frac{4 \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 8.3.2.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\sec (θ) = \frac{4 \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 8.3.2.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$\sec (θ) = \frac{4 \sqrt{7}}{7^{\frac{2}{2}}}$

Step 8.3.2.6.4

Cancel the common factor of

$2$ .

Tap for more steps...

Step 8.3.2.6.4.1

Cancel the common factor.

$\sec (θ) = \frac{4 \sqrt{7}}{7^{\frac{2}{2}}}$

Step 8.3.2.6.4.2

Rewrite the expression.

$\sec (θ) = \frac{4 \sqrt{7}}{7}$

Step 8.3.2.6.5

Evaluate the exponent.

$\sec (θ) = \frac{4 \sqrt{7}}{7}$

Step 9

Find the value of cosecant.

Tap for more steps...

Step 9.1

Use the definition of cosecant to find the value of

$\csc (θ)$ .

$\csc (θ) = \frac{hyp}{opp}$

Step 9.2

Substitute in the known values.

$\csc (θ) = \frac{4}{3}$

Step 10

This is the solution to each trig value.

$\sin (θ) = \frac{3}{4}$

$\cos (θ) = \frac{\sqrt{7}}{4}$

$\tan (θ) = \frac{3 \sqrt{7}}{7}$

$\cot (θ) = \frac{\sqrt{7}}{3}$

$\sec (θ) = \frac{4 \sqrt{7}}{7}$

$\csc (θ) = \frac{4}{3}$