Trigonometric Functions

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Pre-Calculus › Trigonometric Functions

Questions 1 - 10

Find the length of an arc of a circle if the radius is and the angle is $\frac{\pi}{4}$ radians.

$\frac{3\pi}{4}$

$\frac{5\pi}{4}$

$\frac{3\pi}{2}$

$\frac{\pi}{4}$

$\frac{\pi}{2}$

Explanation

Write the formula to find the arc length given the angle in radians.

$s=r\theta$

Substitute the radius and angle.

$s=3\left(\frac{\pi}{4}\right)=\frac{3\pi}{4}$

Convert $660^{\circ}$ to radians.

$\frac{11\pi }{3}$

$\frac{66\pi }{7}$

$\frac{13\pi }{3}$

$\frac{8\pi }{3}$

$\frac{111\pi }{3}$

Explanation

To convert from degrees to radians, you must multiply by $\frac{\pi }{180}$ : $660\cdot \frac{\pi }{180}$ . Then, multiply across as you do normally with fractions. Try and simplify if you can. In this case, 60 goes into both 660 and 180.

Thus, your answer is $\frac{11\pi }{3}$ .

Convert $660^{\circ}$ to radians.

$\frac{11\pi }{3}$

$\frac{66\pi }{7}$

$\frac{13\pi }{3}$

$\frac{8\pi }{3}$

$\frac{111\pi }{3}$

Explanation

Thus, your answer is $\frac{11\pi }{3}$ .

Determine the value of $225^\circ$ in radians.

$\frac{5\pi}{4}rad$

$\frac{40500}{\pi}rad$

$\frac{3\pi}{4}rad$

$\frac{5\pi}{3}rad$

Explanation

To convert from degrees to radians, you do:

$225deg\cdot \frac{(\pi) rad}{180 deg}= \frac{(45\cdot5)\cdot\pi}{45\cdot4} rad= \frac{5\pi}{4} rad$

Determine the value of $225^\circ$ in radians.

$\frac{5\pi}{4}rad$

$\frac{40500}{\pi}rad$

$\frac{3\pi}{4}rad$

$\frac{5\pi}{3}rad$

Explanation

To convert from degrees to radians, you do:

$225deg\cdot \frac{(\pi) rad}{180 deg}= \frac{(45\cdot5)\cdot\pi}{45\cdot4} rad= \frac{5\pi}{4} rad$

Which of the following is equivalent to the expression:

$\frac{1-sin^2(x)}{csc^2(x)}$

Explanation

Which of the following is equivalent to the following expression?

$\frac{1-sin^2(x)}{csc^2(x)}$

Recall our Pythagorean trig identity:

It can be rearranged to look just like our numerator:

So go ahead and change our original expression to:

$\frac{cos^2(x)}{csc^2(x)}$

Then recall the definition of cosecant:

$csc(x)=\frac{1}{sin(x)}$

So our original expression can be rewritten as:

$cos^2(x)\div\frac{1}{sin^2(x)}=cos^2(x)sin^2(x)$

So our answer is:

Find the length of an arc of a circle if the radius is and the angle is $\frac{\pi}{4}$ radians.

$\frac{3\pi}{4}$

$\frac{5\pi}{4}$

$\frac{3\pi}{2}$

$\frac{\pi}{4}$

$\frac{\pi}{2}$

Explanation

Write the formula to find the arc length given the angle in radians.

$s=r\theta$

Substitute the radius and angle.

$s=3\left(\frac{\pi}{4}\right)=\frac{3\pi}{4}$

Find the length of an arc of a circle if the radius is and the angle is $\frac{\pi}{4}$ radians.

$\frac{3\pi}{4}$

$\frac{5\pi}{4}$

$\frac{3\pi}{2}$

$\frac{\pi}{4}$

$\frac{\pi}{2}$

Explanation

Write the formula to find the arc length given the angle in radians.

$s=r\theta$

Substitute the radius and angle.

$s=3\left(\frac{\pi}{4}\right)=\frac{3\pi}{4}$

Find the exact value of each expression below without the aid of a calculator.

$sin165^{\circ}$

$\pm\sqrt\frac{\left(1-\frac{\sqrt(3)}{2}\right)}{2}$

$\pm\sqrt\frac{\left(1-\frac{1}{2}\right)}{2}$

$\pm\sqrt\frac{\left(1+\frac{\sqrt(3)}{2}\right)}{2}$

$\pm\sqrt\frac{(1+\frac{1}{2})}{2}$

$\pm1$

Explanation

In order to find the exact value of $\sin165^\circ$ we can use the half angle formula for sin, which is

$sin\left(\frac{\alpha}{2}\right) = \pm\sqrt\frac{1-cos\alpha}{2}$ .

This way we can plug in a value for alpha for which we know the exact value. $165^\circ$ is equal to $330^\circ$ divided by two, and so we can plug in $330^\circ$ for the alpha above.