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

Even and Odd Functions

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

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

Simplifying Using Identities

Simplify \frac{sin\theta}{cos\theta}

1. The above equation can be split into two components

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

2. Referring to the list of trig identities, we know that \frac{1}{csc\theta} is equal to sin\theta.

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

3. Simplifying further, we get:

 sin^2\theta 

Finding all values using identities

If sec\theta = -\frac{25}{7} and 0 < \theta < \pi, find the values of the other 5 trig functions:

1. To find tan\theta, we can use the trig identity 1 + tan^2\theta = sec^2\theta:

 1 + tan^2\theta = (-\frac{25}{7})^2 

Shuffling things around, we get this:

 tan^2\theta = \frac{625}{49} - 1 

Performing that subtraction gives us this:

 \frac{625}{49} - \frac{49}{49} = \frac{576}{49} = tan^2\theta 

You can get rid of the exponent:

 \sqrt{\frac{576}{49}} = tan\theta 

\sqrt{576} = 24 and \sqrt{49} = 7, so:

 tan\theta = \frac{24}{7} 

2. To find cos\theta, because sec is the inverse of cos, we can use the identity sec\theta = \frac{1}{cos\theta}: cos\theta = -\frac{7}{25}

3. To find sin\theta, we can use the trig identity sin^2\theta + cos^2\theta = 1:

 sin^2\theta + (-\frac{7}{25}) = 1 

Rearranging, we get:

 1 - (-\frac{7}{25})^2 = sin^2\theta 

Applying the exponent gives us \frac{49}{625}, so we can do this:

 \frac{625}{625} - \frac{49}{625} = \frac{576}{625} = sin^2\theta 

Getting rid of the exponent:

 \sqrt{\frac{576}{625}} = \frac{24}{25} = sin\theta 

From there, you can find the rest of the identities fairly easily.