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

An identity is an equation that is true for all values of the variable for which the expressions in the equation are defined.

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}}

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.

Simplifying trig expressions using identities

Given the difference of square formula:

 a^2 - b^2 = (a-b)(a+b) 

Examples

Simplify \tan\theta\sin\theta + \cos\theta:

1. \dfrac{\sin\theta}{\cos\theta} * \sin\theta + \cos\theta
2. \dfrac{\sin^2\theta}{cos\theta} + \cos\theta
3. (\dfrac{\sin^2\theta}{cos\theta} + \cos\theta)\dfrac{\cos\theta}{\cos\theta} = \sin^2\theta*\cos^2\theta + \cos\theta

Simplify $\defrac{2 + tan^2}