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## SAT (Fall 2023)

### Course: SAT (Fall 2023)>Unit 6

Lesson 6: Additional Topics in Math: lessons by skill

# Angles, arc lengths, and trig functions | Lesson

## What are "angles, arc lengths, and trig functions" problems, and how frequently do they appear on the test?

Note: On your official SAT, you'll likely see at most 1 question that tests your knowledge of the skills we teach in this article. Make sure you understand the more frequently-tested skills on the SAT before you spend time practicing this skill.

The problems in this lesson involve circles and angle measures in radians, a unit for angle measure much like degrees. We can use radian measures to calculate arc lengths and sector areas, and we can calculate the sine, cosine, and tangent of radian measures.
In this lesson, we'll learn to:
1. Convert between radians and degrees
2. Use our knowledge of special right triangles to find radian measures
3. Identify the sine, cosine, and tangent of common radian measures
This lesson builds upon the following skills:
You can learn anything. Let's do this!

## How do I convert between radians and degrees?

### Converting between radians and degrees

At the beginning of each SAT math section, the following information about radians and degrees is provided as reference:
• The number of degrees of arc in a circle is $360$.
• The number of radians of arc in a circle is $2\pi$.
This means $360$ degrees is equivalent to $2\pi$ radians, and $180$ degrees is equivalent to $\pi$ radians. We can set up a proportional relationship to convert between radian and degree measures.
$\frac{\text{radian measure}}{\pi }=\frac{\text{degree measure}}{{180}^{\circ }}$

Example: Convert ${90}^{\circ }$ to radians.

This also means we can use radian measures to calculate arc lengths and sector areas just like we can with degree measures:
$\frac{\text{central angle}}{2\pi }=\frac{\text{arc length}}{\text{circumference}}=\frac{\text{sector area}}{\text{circle area}}$

Example: In a circle with center $O$, central angle $AOB$ has a measure of $\frac{2\pi }{3}$ radians. The area of the sector formed by central angle $AOB$ is what fraction of the area of the circle?

### Try it!

try: compare radian and degree measures
Order the following angle measures from smallest to largest.

## How do I use special right triangles to find radian measures?

Note: The topics covered in this section have not appeared in recent SATs, but they could in the future! If they do, it will likely only be 1 question.

### Trig values of special angles

Trig values of π/4See video transcript

### Special right triangles in circles

At the beginning of each SAT math section, the following information about special right triangles is provided as reference:
These angle measures and their radian equivalents appear frequently in questions about circles and circle trigonometry. The table below shows the angles in special right triangles and their equivalent radian measures.
${30}^{\circ }$$\frac{\pi }{6}$
${45}^{\circ }$$\frac{\pi }{4}$
${60}^{\circ }$$\frac{\pi }{3}$
The radian measures we'll see on the SAT are usually multiples of the ones shown above.
On the test, we may be asked to find the radian measure of a central angle in a circle in the $xy$-plane, such as that of angle $AOB$ in the figure below. To do so, we'll draw a right triangle and look for the side length relationships in the special right triangles above.
We can draw a right triangle using the radius $\stackrel{―}{OA}$ as the hypotenuse. Since one vertex of the right triangle is the origin, the two legs of the right triangle have lengths equal to the $x$- and $y$- coordinates of point $A$.
Since the two legs of the right triangle have the same length, we can conclude that it is a $45$-$45$-$90$ special right triangle, and the measure of angle $AOB$ must be ${45}^{\circ }$ or $\frac{\pi }{4}$ radians.

### Try it!

try: recognize a special right triangle in a circle
In the figure above, $O$ is the center of a circle in the $xy$-plane. The measure of angle $AOB$ is $\frac{\pi }{6}$ radians.
If the $x$-coordinate of point $A$ is $2\sqrt{3}$, what is its $y$-coordinate?
What is the radius of the circle?

## How do I find the sine, cosine, and tangent of radian measures?

Note: The topics covered in this section have not appeared on recent tests, but they could show up on future tests! Questions on trigonometry in radians are a rare variety of an already infrequently-tested skill.

### The trig functions & right triangle trig ratios

The trig functions & right triangle trig ratiosSee video transcript

Trigonometry using radian measures is based on the unit circle, a circle centered on the origin with a radius of $1$.
We can describe each point $\left(x,y\right)$ on the circle and the slope of any radius in terms of $\theta$:
• $x=r\mathrm{cos}\theta =\mathrm{cos}\theta$
• $y=r\mathrm{sin}\theta =\mathrm{sin}\theta$
• $\frac{y}{x}=\mathrm{tan}\theta$
The table below shows the sine, cosine, and tangent of some common radian measures in the unit circle:
Note: If you already know these, that's great! If not, consider spending time on the more frequently-tested skills on the SAT before familiarizing yourself with the values of trigonometric functions.
$\theta$$x$ or $\mathrm{cos}\theta$$y$ or $\mathrm{sin}\theta$$\mathrm{tan}\theta$
$0$$1$$0$$0$
$\frac{\pi }{6}$$\frac{\sqrt{3}}{2}$$\frac{1}{2}$$\frac{\sqrt{3}}{3}$
$\frac{\pi }{4}$$\frac{\sqrt{2}}{2}$$\frac{\sqrt{2}}{2}$$1$
$\frac{\pi }{3}$$\frac{1}{2}$$\frac{\sqrt{3}}{2}$$\sqrt{3}$
$\frac{\pi }{2}$$0$$1$undefined
$\frac{2\pi }{3}$$-\frac{1}{2}$$\frac{\sqrt{3}}{2}$$-\sqrt{3}$
$\frac{3\pi }{4}$$-\frac{\sqrt{2}}{2}$$\frac{\sqrt{2}}{2}$$-1$
$\pi$$-1$$0$$0$

The number of radians in a $135$-degree angle can be written as $a\pi$, where $a$ is a constant. What is the value of $a$ ?

practice: use special right triangle to find radian measure
In the $xy$-plane above, $O$ is the center of the circle, and the measure of $\mathrm{\angle }AOB$ is $\frac{\pi }{a}$. What is the value of $a$ ?

## Things to remember

$\frac{\text{radian measure}}{\pi }=\frac{\text{degree measure}}{{180}^{\circ }}$
We can describe each point $\left(x,y\right)$ on the unit circle and the slope of any radius in terms of $\theta$:
• $x=\mathrm{cos}\theta$
• $y=\mathrm{sin}\theta$
• $\frac{y}{x}=\mathrm{tan}\theta$

## Want to join the conversation?

• what does the symbol θ represent
• It is just a substitute for an angle you don’t know. Kind of like variables.
• I'm a little bit confused...Where in the SAT math courses on Khan Academy do I find more about it?
(1 vote)
• If you want more practice on this, you can go to the practice tab in your SAT dashboard and scroll down in the math section until you get to "Angles, arc lengths, and trig functions" under Additional Topics in math. If you want some more learning, check out some videos in Khan's high school geometry course, outside of the SAT section. Did that answer your question?
• So, I'm alright if I'm using a calculator on these problems. However, it's mainly in the No Calculator section. How would I go about doing this (especially in the use of pi), besides just being fantastic at mental math? Thanks.
• I would just advise you to practice doing SAT non calc problems. The more practice you do, the better you will get at doing the mental math faster and more efficiently. Hope this helps!
• Is there a pattern or a way to intuitively understand the values of trigonometric functions or is memorization the only approach to remember the sine, cosine, and tangent of some common radian measures in the unit circle?
• Well you can check the derivations for how we got these values, maybe that might help? The values are now embedded in my mind from practice and memorization.
BUT since you were asking for a trick/pattern to remember the table, here's something my tenth grade math teacher taught us (though I've never had the need to use it):
^ First remember the order of the angles: 0, pi/6, pi/4, pi/3 and pi/2
^ For the sin table, write the numbers 0, 1, 2, 3, 4 under each angle respectively
^ Now take the square root of the numbers and THEN divide them by 2
You'll be left with:
Angle Sintheta
0 0
pi/6 1/2
pi/4 rt.2/2
pi/3 rt.3/2
pi/2 1

^ Now for cosx it's just the other way around: 1, rt.3/2 etc
^ For tan it's just sin/cos
Hope this helps!!