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## Trigonometry

### Course: Trigonometry>Unit 1

Lesson 5: Sine and cosine of complementary angles

# Intro to the Pythagorean trig identity

Sal introduces and proves the identity (sinθ)^2+(cosθ)^2=1, which arises from the Pythagorean theorem! Created by Sal Khan.

## Want to join the conversation?

• Are arccos and secant the same?
• To add to Steven's excellent answer, let note a point at which beginning student get confused.
The inverses of the trig function are indicated either with arc- or with ⁻¹. However, the inverses are NOT the reciprocal of the trig functions. The confusion come from the fact that ⁻¹ is sometimes used to indicate a reciprocal. So, the problem is an inconsistency of notation that has, unfortunately, become standard.
Thus, cos⁻¹ x = arccos x. It does NOT equal sec x or 1 / cos x
The same goes for all of the trigonometric functions and their inverses.

To make it more confusing (cos x)⁻¹ does mean sec x. So, the ⁻¹ only means the inverse function instead of the reciprocal when it is written immediately after the name of the function but before the argument.
• At , he says "sin^2(theta)" why didn't he say "sin(theta)^2"?
• sin ^2 theta is the more "formal" way to write out the equation, as it avoids the possibility of mistaking the theta as "theta^2".
• So I'm doing trig in foundations 11 and we are just reviewing grade 10. In grade 10 I completely understood but now I find it harder than ever and I looked to you for help but I don't know what "THEDA" is ?
• Theta is a Greek letter. It usually represents an unknown angle measure. So...I can literally say...I don't know what Theta is either. Oh I'm gonna die laughing. That was good right? Kidding. Yeah, Theta represents the unknown angle measure, it's like a variable. :)
• Sal writes sin²Θ + cos²Θ = 1, shouldn't it be (sinΘ)² + (cosΘ)² = 1?
• Those mean the same thing. However, it is customary to write it as sin² Θ instead of (sin Θ )² to avoid confusion with sin (Θ²).
• At , it says sin^2 θ+cos^2 θ=1.
So does it mean that sin θ=(1-cos^2 θ)^1/2 and cos θ=(1-sin^2 θ)^1/2?
• At right? yes, it is true that sin θ=(1-cos^2 θ)^1/2 and cos θ=(1-sin^2 θ)^1/2, we can achieve that by simple rearrangement of the equation.
• Does this identity apply to non-right triangles as well?
• No, it doesn't. This identity uses a particular property with right triangles called the Pythagorean theorem, where the hypotenuse's length is equal to the square root of the sum of the squares of the legs of the triangle. Plus, the trigonometric ratios themselves apply to right triangles. There are things that apply to all triangles, but I don't want to explain them in this reply because it may be confusing.
(1 vote)
• Is sin^2(theta) equal to the sin(theta), and then all of that squared? Or is it theta squared, and then the sine of that? I'm guessing it's the former, but just making sure. . .
• no SIn^2(theta ) is sin squared and then multiplied by theta
• Why is cos(2x) = 1-2sin^2x ?
• sin and cos are complementary, look at Sine and cosine of complementary angles in trigonometry
(1 vote)
• When you write "sine squared theta," is that a new property of the sine function (like arcsine, which looks like sine to the negative one power), or are you squaring theta? Are you really squaring the sine or squaring the value (theta)?
• You're squaring the sin, not squaring theta. You take the sin of theta first, then square that.
Example-
sin^2(pi/4) = (sqrt2/2)^2 = 1/2