If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Course: Precalculus (Eureka Math/EngageNY)>Unit 4

Lesson 2: Topic A: Lessons 3-4: Addition and subtraction formulas

Finding trig values using angle addition identities

Sal finds the value of sin(7π/12) by rewriting it as sin(π/3+π/4) and then using the sine angle addition formula. Created by Sal Khan.

Want to join the conversation?

• If the angle cannot be decomposed into pi/4, pi/6 and pi/3 is there a way to find the sine without using a calculator? For example 7pi/11.
• You can use the Taylor Series for sin centered at 0:
x - (x^3)/3! + (x^5)/5! - (x^7)/7! + (x^9)/9! - (x^11)/11!, etc.
Plugging in 7pi/11, you get:
7pi/11 - ((7pi/11)^3)/3! + ((7pi/11)^5)/5!, etc.
It's a lot of work for 7pi/11, but it still works.
Hope this helps!
• So which videos cover the angle addition identities? I'm confused. I never saw any of the videos he references at the beginning of this one that cover the angle addition identities, and can't seem to find them upon searching the site. Furthermore, in the following quiz, there are questions regarding tangent addition identities, which aren't covered in any video I've seen yet on this site, let alone this one. If anyone could point me in the right direction, that would be great.
• In the video, couldn't the magenta line be refereed to as the radius since it's extending from the center. I understand what this video is saying. But I was just curious why the magenta line isn't 1.
• It's extending from the center, but it doesn't reach the circle itself. It's almost the length of the radius, but not quite.
• Edit: I know this question was 3 years ago. At that time I was just relearning math and couldn't see it. Now looking back at this question. It was so simple. All we need to do is factor out √(3) from the numerator and denominator.

------------------------------------------------------------
Original question:

How do you convert (3-√(3)) / (3+√(3)) to (√(3)-1) / (√(3)+1)?
My approach was using tan(45°-30°) which led to the answer (3-√(3))/(3+√(3)).

The answer choice to the problem is (√(3)-1) / (√(3)+1) by using the approach tan(60°-45°).

I plugged both answers into the calculator, both gave the exact same value of 0.267949192. Which means they are same answer written in different forms. So how do you algebraically manipulate so that one transform into another?
• You factorise the square root of 3 out of the first equation from the numerator and denominator to get root 3 divided by root 3 and then multiplied by the second equation. The root of 3 divided by itself is just 1 and thus you are left with the second equation alone...
• Can this technique be used for every angle?
• Not quite. The angles that they're picking are ones that can be made by adding angles that are easy to remember, namely pi/6, pi/4, pi/3, and pi/2 (30, 45, 60, and 90, respectively) and their multiples. You can use angle addition to quickly find the trig values of, say, 75 degrees, since it's easy to see that 45+30=75.

However, if you are trying to find the trig value of, say, 33 degrees, the angle addition identities won't help you much since you can't add or subtract any of the numbers mentioned above to get to 33. In short, angle addition identities are only good for figuring quick trig values if the angle can be added to (or subtracted to) using the angles mentioned above, or multiples of them.
• I am having problems deciding whether to use addition or subtraction formula with larger angles. For example, if the problem asks for cos(165) why can't I add 135 and 30? The problem suggested 225-60. The hints do not explain how to choose the correct formula. I am very confused with this please help!!
• as long as your sum equals 165, it should work. there are multiple ways to rewrite 165 using the "known angles" (30, 45, 60, 90, 120, 135...), but they all end up the same. if you check 135+30 and 225-60, they are the same thing.
• How do you know what the angle looks like at ? I get lost round that part of the video.
• You need to become more familiar with the unit circle and radians.

Radians are actually easier to work with than degrees.
The video mentioned 7π/12, right?
Rather than think of the circles as having 360 degrees, it has 2π radians.
That means π is 180 degrees, and π/2 is 90 degrees.
Now 7/12 is just a bit more than 6/12, and 6/12=1/2, right?
So 7π/12 is just a wee bit more than 6π/12, which is π/2 which is 90 degrees - so 7π/12 is just a wee bit more than 90 degrees.

It might seem complicated at first, but it is way easier, especially when you get into higher math, where degrees are never used.

Try these:
• Doesn't a 45 45 90 triangle have legs with a length of 1/sqrt(2)
• Not all 45-45-90 triangles are congruent, so they won't all have the same side lengths. As far as the standard reference triangle, some people think that the sides are 1/sqrt(2) - 1/sqrt(2) - 1, and some people think that it is 1 - 1 - sqrt(2). Those are both similar right triangles with 45 degree acute angles, and the trig and other math will work out on them in exactly the same way.
• Does cos(3pi/4)=-sqrt(2)/2 or +sqrt(2)/2?

Thanks.