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Course: Digital SAT Math > Unit 9
Lesson 5: Unit circle trigonometry: mediumUnit circle trigonometry | Lesson
A guide to unit circle trigonometry on the digital SAT
What are unit circle trigonometry problems?
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:
- Convert between radians and degrees
- Use our knowledge of special right triangles to find radian measures
- Identify the sine, cosine, and tangent of common radian measures
This lesson builds upon the following skills:
- Right triangle trigonometry
- Circle theorems
You can learn anything. Let's do this!
How do I convert between radians and degrees?
Radians & degrees
Converting between radians and degrees
To convert between radians and degrees, we must be aware of the following information:
- The number of degrees of arc in a circle is
. - The number of radians of arc in a circle is
.
This means degrees is equivalent to radians, and degrees is equivalent to radians. We can set up a proportional relationship to convert between radian and degree measures.
Example: Convert to radians.
This also means we can use radian measures to calculate arc lengths and sector areas just like we can with degree measures:
Example: In a circle with center , central angle has a measure of radians. The area of the sector formed by central angle is what fraction of the area of the circle?
Try it!
How do I use special right triangles to find radian measures?
Trig values of special angles
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.
Degree measure | Radian measure |
---|---|
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 -plane, such as that of angle 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 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 - and - coordinates of point .
Since the two legs of the right triangle have the same length, we can conclude that it is a - - special right triangle, and the measure of angle must be or radians.
Try it!
How do I find the sine, cosine, and tangent of radian measures?
Unit circle definition of trig functions
The trig functions & right triangle trig ratios
Trigonometry using radian measures
Trigonometry using radian measures is based on the unit circle, a circle centered on the origin with a radius of .
We can describe each point on the circle and the slope of any radius in terms of :
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.
undefined | |||
Your turn!
Things to remember
We can describe each point on the unit circle and the slope of any radius in terms of :
Want to join the conversation?
- You guys i have a trick that i was taught in school and i might as well share it, hope you find it helpful
If you are converting to degrees, always multiply by 180(because its in degrees) then divide by pi
if you are converting to rad however just multiply by pi(because it in rad) then divide by 180
so lets say they ask to convert 5pi/6 to degrees, just multiply by 180(because its degrees you are converting to) then divide by pi. This way you dont have to waste time overthinking it.
This advice is for those who still find this concept confusing. Hope i helps and good lick with your DSAT(165 votes)- Your tip was really helpful.Thank you(6 votes)
- bruh...I am never gonna pass this levelllllll(30 votes)
- Trigonometry is not that hard. In fact, it is one of my favorite parts in Math. It is important to do regular practice to be good at trigo. Start with easier sums, and slowly increase your difficulty level. You will begin to like it!(49 votes)
- i haven't even learned this yet in school 💀 rip(50 votes)
- idk if this'll help at all but just memorize the special triangles, keep soh cah toa in mind, and basic triangle knowledge (ie. the pythagorean theorem, the sum of angles of a triangle, etc.) and you should be able to solve everything logically from there if you're taking the sat :p(7 votes)
- im in tears. i just want a good score.(29 votes)
- Dry your tears + study at Khan = good score(16 votes)
- I did not learn anything from this method.(19 votes)
- Does anyone know where I can get practice apart from Khan Academy?(8 votes)
- Download the bluebook from the CollegeBoard website, in it are 4 digital SAT practice tests:)
Download it here:
https://bluebook.app.collegeboard.org/(11 votes)
- Do I have to finish the medium level lesions or should I just do the practice tests and revise?My exam is on 26th Aug(12 votes)
- is this something common in the actual exam?(9 votes)
- I only came across two question one in a practice test and the other was in the paper trial in december 2022(7 votes)
- Is khan academy enough to score 1350+?(7 votes)
- for math you can easily achieve 700+ from KA, but for english use some other sources too(3 votes)
- This was good(9 votes)