# Trigonometric Equations

## Trigonometric Equations Revision

**Solving Trigonometric Equations**

Using what we know already about **sine**, **cosine**, and **tangent**, (or **sin**, **cos**, and **tan**), we need to be able to solve equations involving these.

For example,

\sin(x)=\dfrac{\sqrt{3}}{2}

We will look at how to solve equations like this **graphically**, and **algebraically**.

**Solving Graphically**

We can look at the **sin**, **cos**, and **tan** graphs to solve equations.

For example, estimate the solutions to

\sin(x)=0.2

\cos(x)=0.2

\tan(x)=0.2

Between 0\degree and 360\degree

**Sine graph**

To solve, draw a line on the sine graph at y=0.2 and read off the solutions:

The yellow line crosses the **sin** graph at approximately x=11\degree and x=169\degree. These are the approximate solutions to the equation \sin(x)=0.2

**Cosine**** graph**

To solve, draw a line on the cosine graph at y=0.2 and read off the solutions:

The yellow line crosses the **cos** graph at approximately x=78\degree and x=282\degree. These are the approximate solutions to the equation \cos(x)=0.2

**Tangent graph**

To solve, draw a line on the tangent graph at y=0.2 and read off the solutions:

The yellow line crosses the **tan** graph at approximately x=11\degree and x=191\degree. These are the approximate solutions to the equation \tan(x)=0.2

**Solving Algebraically – Sine**

Sometimes, you may not be given a graph, and so, it would not be possible to accurately solve a trigonometric equation graphically. In this case, we would solve the equation algebraically, using our calculator.

You will always be asked to solve algebraic trigonometric equations for values between 0\degree and 360\degree.

We can use our calculator, and the properties of **sine** to solve trigonometric equations involving **sine**.

For example:

\sin(x)=\dfrac{\sqrt{3}}{2}

We can use our **calculator**:

\sin^{-1}\left(\dfrac{\sqrt{3}}{2}\right)=60\degree

However, our calculator only gives us** one value**, but we know there are more solutions between 0\degree and 360\degree.

As the **sine** graph is positive between 0\degree and 180\degree, we can find the other value by…

180\degree-60\degree=120\degree

So the final solutions are x=60\degree, 120\degree.

**Solving Algebraically – Cosine**

We can use our calculator, and the properties of **cosine** to solve trigonometric equations involving **cos**.

For example:

\cos(x)=\dfrac{1}{2}

We can use our **calculator**:

\cos^{-1}\left(\dfrac{1}{2}\right)=60\degree

However, our calculator only gives us** one value**, but we know there are more solutions between 0\degree and 360\degree.

As the **cosine** graph is positive between 0\degree and 360\degree, we can find the other value by…

360\degree-60\degree=300\degree

So the final solutions are x=60\degree, 300\degree.

**Solving Algebraically – Tangent**

We can use our calculator, and the properties of **tangent** to solve trigonometric equations involving **tan**.

For example:

\tan(x)=\sqrt{3}

We can use our **calculator**:

\tan^{-1}(x)=60\degree

However, our calculator only gives us** one value**, but we know there are more solutions between 0\degree and 360\degree.

As the **tangent** graph repeats every 180\degree, we can find the other value by…

60\degree+180\degree=240\degree

So the final solutions are x=60\degree, 240\degree.

**Example 1: Solving Graphically**

Use the **cosine** graph to estimate the solutions to

\cos(x)=0.4

between 0\degree and 360\degree

**[3 marks]**

Draw the line y=0.4 onto the graph and read off the solutions:

The solutions are approximately x=68\degree and x=293\degree

**Example 2: Solving Algebraically**

Solve

\sin(x)=\dfrac{5}{8}

For values between 0\degree and 360\degree, giving your answers to 2 decimal places.

**[3 marks]**

Let’s find the first value using our calculator

\sin^{-1}\left(\dfrac{5}{8}\right)=38.68\degree

As **sin** is positive between 0\degree and 180\degree

180-38.68=141.32\degree

Solutions: x=38.68\degree, 141.32\degree

## Trigonometric Equations Example Questions

**Question 1: **Graphically solve the equation

\tan(x)=1.3

approximately, between 0\degree and 360\degree

**[3 marks]**

Draw the line y=1.3 and read the solutions:

Solutions are approximately x=55\degree\text{, }235\degree

**Question 2: **Solve \tan(x)=\dfrac{4}{7} between 0\degree and 360\degree, giving your answer(s) to the nearest whole number.

**[3 marks]**

Using a calculator:

\tan^{-1}\left(\dfrac{4}{7}\right)=30\degreeFind the other value:

30+180=210\degree**Question 3: **Given that \cos^{-1}(x)=45\degree, find another solution to \cos^{-1}(x) between 0\degree and 360\degree, and work out the value of x.

**[4 marks]**

As we are solving for cos, the other value will be

360-45=315\degreeWe can now do the inverse of \cos^{-1}(x) to work out the value of x

\cos(45)=\dfrac{\sqrt2}{2}