Find the Other Trig Values in Quadrant I cos(x)=3/( square root of 13)

$\cos (x) = \frac{3}{\sqrt{13}}$

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.

$\cos (x) = \frac{adjacent}{hypotenuse}$

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.

$Opposite = \sqrt{{(\sqrt{13})}^{2} - {(3)}^{2}}$

Step 4

Simplify inside the radical.

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

Opposite $= \sqrt{{(13^{\frac{1}{2}})}^{2} - {(3)}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

Opposite $= \sqrt{13^{\frac{1}{2} \cdot 2} - {(3)}^{2}}$

Combine $\frac{1}{2}$ and $2$ .

Opposite $= \sqrt{13^{\frac{2}{2}} - {(3)}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

Opposite $= \sqrt{13^{\frac{2}{2}} - {(3)}^{2}}$

Rewrite the expression.

Opposite $= \sqrt{13 - {(3)}^{2}}$

Evaluate the exponent.

Opposite $= \sqrt{13 - {(3)}^{2}}$

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

Opposite $= \sqrt{13 - 1 \cdot 9}$

Multiply $- 1$ by $9$ .

Opposite $= \sqrt{13 - 9}$

Subtract $9$ from $13$ .

Opposite $= \sqrt{4}$

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

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

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

Opposite $= 2$

Step 5

Find the value of sine.

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

$\sin (x) = \frac{opp}{hyp}$

Substitute in the known values.

$\sin (x) = \frac{2}{\sqrt{13}}$

Simplify the value of $\sin (x)$ .

Multiply $\frac{2}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\sin (x) = \frac{2}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}}$

Combine and simplify the denominator.

Multiply $\frac{2}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\sin (x) = \frac{2 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

Raise $\sqrt{13}$ to the power of $1$ .

$\sin (x) = \frac{2 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

Raise $\sqrt{13}$ to the power of $1$ .

$\sin (x) = \frac{2 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (x) = \frac{2 \sqrt{13}}{{\sqrt{13}}^{1 + 1}}$

Add $1$ and $1$ .

$\sin (x) = \frac{2 \sqrt{13}}{{\sqrt{13}}^{2}}$

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\sin (x) = \frac{2 \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sin (x) = \frac{2 \sqrt{13}}{13^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\sin (x) = \frac{2 \sqrt{13}}{13^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sin (x) = \frac{2 \sqrt{13}}{13^{\frac{2}{2}}}$

Rewrite the expression.

$\sin (x) = \frac{2 \sqrt{13}}{13}$

Evaluate the exponent.

$\sin (x) = \frac{2 \sqrt{13}}{13}$

Step 6

Simplify the value of cosine.

Multiply $\frac{3}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\cos (x) = \frac{3}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}}$

Combine and simplify the denominator.

Multiply $\frac{3}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\cos (x) = \frac{3 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

Raise $\sqrt{13}$ to the power of $1$ .

$\cos (x) = \frac{3 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

Raise $\sqrt{13}$ to the power of $1$ .

$\cos (x) = \frac{3 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\cos (x) = \frac{3 \sqrt{13}}{{\sqrt{13}}^{1 + 1}}$

Add $1$ and $1$ .

$\cos (x) = \frac{3 \sqrt{13}}{{\sqrt{13}}^{2}}$

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\cos (x) = \frac{3 \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\cos (x) = \frac{3 \sqrt{13}}{13^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\cos (x) = \frac{3 \sqrt{13}}{13^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\cos (x) = \frac{3 \sqrt{13}}{13^{\frac{2}{2}}}$

Rewrite the expression.

$\cos (x) = \frac{3 \sqrt{13}}{13}$

Evaluate the exponent.

$\cos (x) = \frac{3 \sqrt{13}}{13}$

Step 7

Find the value of tangent.

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

$\tan (x) = \frac{opp}{adj}$

Substitute in the known values.

$\tan (x) = \frac{2}{3}$

Step 8

Find the value of cotangent.

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

$\cot (x) = \frac{adj}{opp}$

Substitute in the known values.

$\cot (x) = \frac{3}{2}$

Step 9

Find the value of secant.

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

$\sec (x) = \frac{hyp}{adj}$

Substitute in the known values.

$\sec (x) = \frac{\sqrt{13}}{3}$

Step 10

Find the value of cosecant.

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

$\csc (x) = \frac{hyp}{opp}$

Substitute in the known values.

$\csc (x) = \frac{\sqrt{13}}{2}$

Step 11

This is the solution to each trig value.

$\sin (x) = \frac{2 \sqrt{13}}{13}$

$\cos (x) = \frac{3 \sqrt{13}}{13}$

$\tan (x) = \frac{2}{3}$

$\cot (x) = \frac{3}{2}$

$\sec (x) = \frac{\sqrt{13}}{3}$

$\csc (x) = \frac{\sqrt{13}}{2}$

Find the Other Trig Values in Quadrant I cos(x)=3/( square root of 13)

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