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Lesson 20: Trigonometric Equations

**Topic**. This lesson covers Chapter 20: Trigonometric Equations.

**WeBWorK**. There is one WeBWorK assignment on today’s material:

`Trigonometry - Equations`

**Lesson Notes**. (download pdf)

**Question of the Day: **If we know $\sin(x)=\frac{1}{2}$, what do we know about the angle $x$?

## Basic Trigonometric Equations

#### Equations of the form tan(x)=c

**Example 20.1**. Solve for $x$: $\tan(x)=\sqrt{3}$

**Observation 20.2**. To solve $\tan(x)=c$, we first determine one solution $x=\tan^{-1}(c).$ Then the general solution is given by:

$$x=\tan^{-1}(c)+n\cdot\pi \text{ where }n = 0, \pm 1, \pm 2, \pm 3, …$$

#### EQUATIONS OF THE FORM cos(x)=c

**Example 20.**4. Solve for $x$: $\cos(x)=\frac{1}{2}$

*VIDEO: Example 20.4, cos(x)=c*

**Observation 20.5**. To solve $\cos(x)=c$, we first determine one solution $x=\cos^{-1}(c)$. Then the general solution is given by:

$$x=\pm\cos^{-1}(c)+2n\cdot\pi \text{ where }n = 0, \pm 1, \pm 2, \pm 3, …$$

#### EQUATIONS OF THE FORM sin(x)=c

**Example 20.7**. Solve for $x$: $\sin(x)=\frac{\sqrt{2}}{2}$

*VIDEO: Example 20.7, sin(x)=c*

**Observation 20.8**. To solve $\sin(x)=c$, we first determine one solution $x=\sin^{-1}(c)$. Then the general solution is given by:

$$x=(-1)^{n}\cdot\sin^{-1}(c)+n\cdot\pi \text{ where }n = 0, \pm 1, \pm 2, \pm 3, …$$

**Summary: **We summarize the different formulas used to solve the basic trigonometric equations in the following table.

Solve: $\sin(x)=c$ | Solve: $\cos(x)=c$ | Solve: $\tan(x)=c$ |

First, find one solution, that is: $\sin^{-1}(c)$. Use: $\sin^{-1}(-c)=-\sin^{-1}(c)$ | First, find one solution, that is: $\cos^{-1}(c)$. Use: $\cos^{-1}(-c)=\pi-\cos^{-1}(c)$ | First, find one solution, that is: $\tan^{-1}(c)$. Use: $\tan^{-1}(-c)=-\tan^{-1}(c)$ |

The general solution is: $x=(-1)^{n} \sin^{-1}(c)+n \pi$ | The general solution is: $x=\pm \cos^{-1}(c)+2 n \pi$ | The general solution is: $x=\tan ^{-1}(c)+n \pi$ |

where $n=0,\pm 1,\pm 2, \ldots$ | where $n=0,\pm 1,\pm 2, \ldots$ | where $n=0,\pm 1,\pm 2, \ldots$ |

**Example 20.10**. Find the general solution of the equation, and state at least $5$ distinct solutions.

a) $\sin(x)=-\frac{1}{2}$

b) $\cos(x)=-\frac{\sqrt{3}}{2}$

## Equations involving trigonometric functions

**Example 20.11**. Solve for $x$

a) $2 \sin (x)-1=0$

b) $\sec (x)=-\sqrt{2}$

c) $7 \cot (x)+3=0$

*VIDEO: Example 20.11 – equations with trig functions (linear)*

**Example 20.12**. Solve for $x$.

a) $\tan ^{2}(x)+2 \tan (x)+1=0 \quad$ b) $2 \cos ^{2}(x)-1=0$

*VIDEO: Example 20.12 – equations with trig functions (quadratic)*

#### Question

$\sin(x)=11$

#### Answer

Here is the answer blah.