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Course: Algebra (all content)>Unit 14

Lesson 12: The inverse trigonometric functions

Intro to arccosine

Sal introduces arccosine, which is the inverse function of cosine, and discusses its principal range. Created by Sal Khan.

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• i am lost. it seems that the only way i can solve these problems is by memorizing the unit circle. can someone please explain the intuition behind this? what happens when the answer is not 30,60,90 or 45 degrees
• Memorizing the unit circle is helpful in Trigonometry but not necessary. I suggest knowing all you can about how the unit circle works. Sal has some great videos on the unit circle that you could watch. When working in radians, most pre-calculus/ trigonometry courses have you work with 30-60-90 triangle and 45-45-90 triangles on the unit circle. Also when your not working with those specific degrees you inquired about, most courses have you switch to a calculator instead of the by hand method.
But anything on the unit circle can be solved with knowing the Pythagorean theorem as Jatin stated already.
• what is the difference between range and domain?
• The domain is the set of all of the values that you can feed into a function, and the range is the set of all of the values that can come out of it.
• I'm having a lot of trouble with this subject. Could somebody walk me through a detailed explanation of this problem; What is the principal value of sin^-1 (-1/2)?
• I can never find a better explanation to a topic than that explained by Sal.
• Is there any set formula that could be used to find arcsin, arccos, and arctan?
• Not really, but it's easy and much better if you just think it through, It just takes an extra few seconds and it ensures you don't make any mistakes.
• Why do you restrict arccos to only the first two quadrants?
• Because in a function F defined as F(x)=cos(x) for example, you cannot have one x giving multiple F(x) values. If you accepted multiple y values to one x value, then it would not be a function, because of the definition of what a function is, but a binary relation instead.
• Could there be such a thing as arcsecant, arccosecant, and arccotangent?
• At , how does Sal know the triangle is a 30-60-90?
• Because of the sides.
The basic 30-60-90 triangle has sides 2, 1, and sqr 3 (Watch "Example: Solving a 30-60-90 triangle", "Intro to 30-60-90 Triangles", "30-60-90 Triangles II"...), you can use them to find out angles and points on graphs, with this question, instead of 1 the side is 1/2, so to find the rest of the sides you simply half all the sides of the basic triangle and it is still a 30-60-90 triangle but now it fits the triangle on the graph and you can solve the problem.
• Is the relationship between arccos and cos the same as the relationship between logarithms and exponents?
• Yes, Arc cosine is the inverse of cosine and vice versa
arccos = cos^-1
cos = arccos^-1
not to be confused with secant which is the reciprocal
• At you explain that theta is the angle that when intersected with the unit circle gives an x value of -1/2. However wouldn't that mean that theta equals 60 instead of 120? Please explain why we chose theta as 120. I'm having difficulty figuring out how to identify which angle is theta.
• Hello Seahawks,

The angles are always measure counter clockwise with respect to the X axis. For this problem think of a clock. Start at 3 o'clock and move the hour hand backwards to 11 o'clock. This is where the 120 degrees came from.

You could work the problem starting at 9 o'clock as you mentioned. You will need to do more mental work to keep things straight...

Regards,

APD