Trigonometry Examples

$\sin (θ) = 1$

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.

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

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.

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

Step 3

Replace the known values in the equation.

$Adjacent = \sqrt{{(1)}^{2} - {(1)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

One to any power is one.

Adjacent $= \sqrt{1 - {(1)}^{2}}$

Step 4.2

One to any power is one.

Adjacent $= \sqrt{1 - 1 \cdot 1}$

Step 4.3

Multiply $- 1$ by $1$ .

Adjacent $= \sqrt{1 - 1}$

Step 4.4

Subtract $1$ from $1$ .

Adjacent $= \sqrt{0}$

Step 4.5

Rewrite $0$ as $0^{2}$ .

Adjacent $= \sqrt{0^{2}}$

Step 4.6

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

Adjacent $= 0$

Step 5

Find the value of cosine.

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Step 5.1

Use the definition of cosine to find the value of $\cos (θ)$ .

$\cos (θ) = \frac{adj}{hyp}$

Step 5.2

Substitute in the known values.

$\cos (θ) = \frac{0}{1}$

Step 5.3

Divide $0$ by $1$ .

$\cos (θ) = 0$

Step 6

Find the value of tangent.

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Step 6.1

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

$\tan (θ) = \frac{opp}{adj}$

Step 6.2

Substitute in the known values.

$\tan (θ) = \frac{1}{0}$

Step 6.3

Division by $0$ results in tangent being undefined at $θ$ .

$\tan (θ) = Undefined$

Undefined

Step 7

Find the value of cotangent.

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Step 7.1

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

$\cot (θ) = \frac{adj}{opp}$

Step 7.2

Substitute in the known values.

$\cot (θ) = \frac{0}{1}$

Step 7.3

Divide $0$ by $1$ .

$\cot (θ) = 0$

Step 8

Find the value of secant.

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Step 8.1

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

$\sec (θ) = \frac{hyp}{adj}$

Step 8.2

Substitute in the known values.

$\sec (θ) = \frac{1}{0}$

Step 8.3

Division by $0$ results in secant being undefined at $θ$ .

$\sec (θ) = Undefined$

Undefined

Step 9

Find the value of cosecant.

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Step 9.1

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

$\csc (θ) = \frac{hyp}{opp}$

Step 9.2

Substitute in the known values.

$\csc (θ) = \frac{1}{1}$

Step 9.3

Divide $1$ by $1$ .

$\csc (θ) = 1$

Step 10

This is the solution to each trig value.

Undefined