The most important identity in trigonometry is the Pythagorean identity. It comes from the Pythagorean theorem and it says that the sum of the squares of sine and cosine of the same angle is always equal to one. You can use the Pythagorean theorem to prove the Pythagorean identity or the other way round. This identity is very useful and can be used to prove some other trigonometric identities. In this lecture you'll also get to practice how to find angle if you are given the cosine and why it is important to also know the quadrant of the angle.
The Pythagorean theorem: The side lengths of a right triangle satisfy a2 + b2 = c2.
The Pythagorean identity: For any angle x, we have sin2x + cos2x = 1.
Use the Pythagorean theorem to prove the Pythagorean identity.
= 0.47 and θ
is in the fourth quadrant, find sinθ
Verify the following trigonometric identity :
Use the Pythagorean identity to prove the Pythagorean theorem.
[5/13] and θ
is in the third quadrant, find cosθ
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Now, 0.47, that's not something I can easily find the square of, so I'll do that on my calculator.0310
0.472 = 0.2209, so that's +0.2209, sin2θ +0.2209 = 1, sin2θ = 1 - 0.2209, which is 0.7791.0317
Sinθ, if we take the square root of both sides, sinθ is equal to plus or minus the square root of 0.7791, which is approximately equal to 0.8827.0360
Now, it's plus or minus because I know that sine squared is this positive number, but I don't know whether this sine is a positive or negative.0382
We're given more information in the problem, θ is in the fourth quadrant.0390
Remember, sine is the y-coordinate, so the sine in the fourth quadrant is going to be negative because the y-coordinate is negative.0395
Because θ is in quadrant 4, sinθ is going to be negative, so we take the negative value, sinθ is approximately equal to -0.8827.0414
The whole key to doing this problem was to start with the Pythagorean identity sin2θ + cos2θ = 1.0446
Once you're given sine or cosine, you could plug those in and figure out the other one except that you can't figure out whether they're positive or negative.0457
Their identity doesn't tell you that so we had to get this little extra information about θ being in the fourth quadrant, that totals that the sinθ is negative and we were able to figure out that it was -0.8827.0463