Trigonometric Equations

The first Ferris wheel was built for the Chicago World's Fair in 1893. It had a diameter of 250 feet and could carry 2160 people in 36 carriages. From the top of the wheel, passengers could see into four states. After loading all the passengers, the wheel made one revolution in nine minutes.

If you are in the bottom carriage of the Ferris wheel at the start of its revolution, your height after [latex]t[/latex] seconds is given by

[latex]h = f(t) = 139 - 125\cos \left(\dfrac{2t}{3}\right)[/latex]

For how long are you more than 240 feet above the ground?

The figure below shows a graph of the height function and a horizontal line at [latex]h = 240{.}[/latex]

From the graph, we see that [latex]h = 240[/latex] at approximately 215 seconds and 325 seconds into the ride. Your height is more than 240 feet between those two times, or for about 110 seconds.

Solving Trigonometric Equations

In the example above, we used a graph to solve the equation [latex]h = 240{,}[/latex] or

[latex]139 - 125\cos \left(\dfrac{2t}{3}\right) = 240[/latex]

To find a more precise solution, we can use algebraic methods. As an example, we'll solve the slightly simpler equation

[latex]139 - 125\cos\theta = 240[/latex]

We'll look for all solutions for [latex]\theta[/latex] between [latex]0°[/latex] and [latex]360°{.}[/latex] We begin by isolating the trigonometric ratio on one side of the equation.

[latex]139 - 125\cos \theta = 240[/latex] Subtract 139 from both sides.}
[latex]- 125\cos \theta = 101[/latex] Divide both sides by -125.
[latex]\cos\theta = -0.808[/latex]

We have solved equations like this one before: we use the inverse cosine to solve for [latex]\theta{.}[/latex] Remember that there are two angles between [latex]0°[/latex] and [latex]360°[/latex] that have a cosine of [latex]-0.808{,}[/latex] one in the second quadrant and one in the third quadrant. The calculator will give us only the second quadrant solution.

[latex]\theta = \cos^{-1}(-0.808) = 143.9°[/latex]

To find the second solution, we need the third-quadrant angle whose cosine is [latex]-0.808{.}[/latex]

Now, the reference angle for [latex]143.9°[/latex] is

[latex]180° - 143.9° = 36.1°[/latex]

and the angle in the third quadrant with the same reference angle is

[latex]180° + 36.1° = 216.1°[/latex]

(See the figure at right.) Thus, the other solution is [latex]216.1°{.}[/latex]

To solve simple equations involving a single trigonometric ratio (either [latex]\sin \theta, \cos \theta,[/latex] or [latex]\tan \theta[/latex]), we can follow the steps below.

We can use a calculator to help us solve equations that do not involve special angles.

Some trigonometric equations have no solution. As we can observe from their graphs or from their definitions, the sine and cosine functions only have values ranging from [latex]-1[/latex] to [latex]1{.}[/latex]

Checkpoint 5.32.

Graphical Solutions

We can use graphs to approximate the solutions to trigonometric equations.

Checkpoint 5.34.

Equations with Squares of Trig Ratios

Simple quadratic equations can be solved by extracting roots. For example, to solve the equation

[latex]4x^2 + 3 = 15[/latex]

we first isolate [latex]x^2{:}[/latex]

[latex]4x^2 = 12\\ x^2 = 3[/latex]

and then take square roots of both sides to find

[latex]x = \pm \\\sqrt{3} \approx \pm 1.732[/latex]

Recall that a quadratic equation may have two real solutions, one (repeated) real solution, or no real solutions. We can use extraction of roots to solve trigonometric equations as well.

Checkpoint 5.36.

Other quadratic equations can be solved by factoring. For example, we can solve the equation

[latex]4x^2 + 4x - 3 = 0[/latex]

by factoring the left side to get

[latex](2x + 3)(2x - 1) = 0[/latex]

Then we apply the Zero Factor Principle to set each factor equal to zero, and solve each equation.

[latex]2x + 3 = 12 2x - 1= 0\\ x = \dfrac{-3}{2} x = \dfrac{1}{2}[/latex]

The solutions are [latex]\dfrac{-3}{2}[/latex] and [latex]\dfrac{1}{2}{.}[/latex]

Snell's Law

When you view an object through a liquid, such as a spoon in a glass of water, or a fish in an aquarium, the object looks distorted or bent. This distortion is caused by refraction of light. Light rays bend when they pass from one medium to another, for instance from water to glass or from glass to air.

A light ray enters the boundary between the two media at a certain angle, called the angle of incidence, but leaves the boundary at a different angle, the angle of refraction. Both angles are acute angles measured from the normal line perpendicular to the boundary, as shown below.

The change of angle is caused by the fact that light travels at different speeds in different media. The relationship between the angle of incidence and the angle of refraction is given by Snell's Law:

[latex]\dfrac{\sin \theta_1}{\sin \theta_2} = \dfrac{v_1}{v_2}[/latex]

where [latex]\theta_1[/latex] is the angle in the medium where light travels at speed [latex]v_1{,}[/latex] and [latex]\theta_2[/latex] is the angle where light travels at speed [latex]v_2{.}[/latex] The ratio of the speeds is called the index of refraction.

Section 5.2 Summary

Vocabulary

• Equation
• Solve
• Zero Factor Principle
• Angle of incidence
• Angle of refraction
• Normal