notes/education/math/MATH1060 (trig)/Identities.md

Trigonometric Identities

All of the following only apply when the denominator is not equal to zero.

 tan \theta = \frac{y}{x} 

Because the following are inverses of their counterparts, you only need to remember the equivalents for sin, cos, and tan, then just find the inverse by taking 1/v.

Base Identity	Inverse Identity	Alternate Identities	Alternate Inverse Identities
sin\theta = y	csc\theta = \frac{1}{y}		csc\theta = \frac{1}{sin\theta}
cos\theta = x	sec \theta = \frac{1}{x}		sec\theta = \frac{1}{cos\theta}
tan\theta = \frac{y}{x}	cot\theta = \frac{x}{y}	tan\theta = \frac{sin\theta}{cos\theta}	cot\theta = \frac{1}{tan\theta} = \frac{cos\theta}{sin{\theta}}

 cot \theta = \frac{x}{y} 

 sec\theta = \frac{1}{cos\theta}

 csc\theta = \frac{1}{sin\theta}

Pythagorean Identities

The Pythagorean identity expresses the Pythagorean theorem in terms of trigonometric functions. It's a basic relation between the sine and cosine functions.

 sin^2 \theta + cos^2 \theta = 1 

There are more forms that are useful, but they can be derived from the above formula:

 1 + tan^2\theta = sec^2\theta 

 cot^2 \theta + 1 = csc^2\theta 

Even and Odd Identities

• A function is even if f(-x) = f(x).
• A function is odd if f(-x) = -f(x)
• Cosine and secant are even
• Sine, tangent, cosecant, and cotangent are odd.

Examples

If cot\theta = -\sqrt{3}, what is cot(-\theta)?

cot is an odd function, and so cot(-\theta) = \sqrt{3}