## Skills Review for Area and Arc Length in Polar Coordinates

### Learning Outcomes

• Identify reference angles for angles measured in both radians and degrees
• Evaluate trigonometric functions using the unit circle
• Solve trigonometric equations
• Apply reduction formulas

In the Area and Arc Length in Polar Coordinates section, rather than use the rectangular coordinate system to calculate area under a curve and arc length, we will use the polar coordinate system. Here we will review how to evaluate sine and cosine functions at specific angle measures, solve trigonometric equations, and use reduction formulas.

## Find Reference Angles

(See Module 7, Skills Review for Polar Coordinates)

## Evaluate Trigonometric Functions Using the Unit Circle

(See Module 7, Skills Review for Polar Coordinates)

## Solve Trigonometric Equations

Trigonometric equations are, as the name implies, equations that involve trigonometric functions. Similar in many ways to solving polynomial equations or rational equations, only specific values of the variable will be solutions, if there are solutions at all. Often we will solve a trigonometric equation over a specified interval. However, just as often, we will be asked to find all possible solutions, and as trigonometric functions are periodic, solutions are repeated within each period. In other words, trigonometric equations may have an infinite number of solutions. Additionally, like rational equations, the domain of the function must be considered before we assume that any solution is valid. The period of both the sine function and the cosine function is $2\pi$. In other words, every $2\pi$ units, the y-values repeat. If we need to find all possible solutions, then we must add $2\pi k$, where $k$ is an integer, to the initial solution. Recall the rule that gives the format for stating all possible solutions for a function where the period is $2\pi :$

$\sin \theta =\sin \left(\theta \pm 2k\pi \right)$

There are similar rules for indicating all possible solutions for the other trigonometric functions. Solving trigonometric equations requires the same techniques as solving algebraic equations. We read the equation from left to right, horizontally, like a sentence. We look for known patterns, factor, find common denominators, and substitute certain expressions with a variable to make solving a more straightforward process. However, with trigonometric equations, we also have the advantage of using the identities we developed in the previous sections.

### Example: Solving a Linear Trigonometric Equation Containing Cosine

Find all possible exact solutions for the equation $\cos \theta =\frac{1}{2}$.

### EXAMPLE: Solving a Linear Trigonometric Equation Containing Sine

Find all possible exact solutions for the equation $\sin t=\frac{1}{2}$.

### EXAMPLE: Solving a Linear Trigonometric Equation Containing Cosine

Solve the equation exactly: $2\cos \theta -3=-5,0\le \theta <2\pi$.

### Try It

Solve exactly the following linear equation on the interval $\left[0,2\pi \right):2\sin x+1=0$.

### Try It

When we are given equations that involve only one of the six trigonometric functions, their solutions involve using algebraic techniques and the unit circle. We need to make several considerations when the equation involves trigonometric functions other than sine and cosine. Problems involving the reciprocals of the primary trigonometric functions need to be viewed from an algebraic perspective. In other words, we will write the reciprocal function, and solve for the angles using the function. Also, an equation involving the tangent function is slightly different from one containing a sine or cosine function. First, as we know, the period of tangent is $\pi$, not $2\pi$. Further, the domain of tangent is all real numbers with the exception of odd integer multiples of $\frac{\pi }{2}$, unless, of course, a problem places its own restrictions on the domain.

### EXAMPLE: Solving a Trigonometric Equation

Solve the problem exactly: $2{\sin }^{2}\theta -1=0,0\le \theta <2\pi$.

### Example: Solving a Trigonometric Equation

Solve the following equation exactly: $\csc \theta =-2,0\le \theta <4\pi$.

### Example: Solving a Trigonometric Equation

Solve the equation exactly: $\tan \left(\theta -\frac{\pi }{2}\right)=1,0\le \theta <2\pi$.

### Try It

Find all solutions for $\tan x=\sqrt{3}$.

### Try It

Sometimes it is not possible to solve a trigonometric equation with identities that have a multiple angle, such as $\sin \left(2x\right)$ or $\cos \left(3x\right)$. When confronted with these equations, recall that $y=\sin \left(2x\right)$ is a horizontal compression by a factor of 2 of the function $y=\sin x$. On an interval of $2\pi$, we can graph two periods of $y=\sin \left(2x\right)$, as opposed to one cycle of $y=\sin x$. This compression of the graph leads us to believe there may be twice as many x-intercepts or solutions to $\sin \left(2x\right)=0$ compared to $\sin x=0$. This information will help us solve the equation.

### Example: Solving a Multiple Angle Trigonometric Equation

Solve exactly: $\cos \left(2x\right)=\frac{1}{2}$ on $\left[0,2\pi \right)$.

## Use Reduction Formulas

### A General Note: Reduction Formulas

The reduction formulas are summarized as follows:

\begin{align}&{\sin }^{2}\theta =\frac{1-\cos \left(2\theta \right)}{2} \\ &{\cos }^{2}\theta =\frac{1+\cos \left(2\theta \right)}{2} \\ &{\tan }^{2}\theta =\frac{1-\cos \left(2\theta \right)}{1+\cos \left(2\theta \right)} \end{align}

### Example: Using reduction formulas to reduce powers

Write an equivalent expression for ${\cos }^{4}x$ that does not involve any powers of sine or cosine greater than 1.

### Example: Using Reduction formulas To Reduce Powers

Use the power-reducing formulas to prove

${\sin }^{3}\left(2x\right)=\left[\frac{1}{2}\sin \left(2x\right)\right]\left[1-\cos \left(4x\right)\right]$

### Try It

Use the power-reducing formulas to prove that $10{\cos }^{4}x=\frac{15}{4}+5\cos \left(2x\right)+\frac{5}{4}\cos \left(4x\right)$.