Solving Trig Equations from Models

Lesson 1 of 2: Isolation and the Principal Solution

In this lesson:

• Isolate the trig expression using familiar algebra
• Apply arcsin, arccos, or arctan to find one solution
• Solve for the original variable using a calculator in radian mode

What You Will Learn Today

By the end of this lesson, you can:

1. Isolate the trig expression in a modeling equation
2. Apply arcsin, arccos, or arctan to find the principal solution
3. Use a calculator in radian mode to evaluate inverse trig values

The Ferris Wheel Model and Question

Model:

= minutes after boarding; = height in feet above ground

Question: When does the rider first reach 40 feet?

What equation do we need to solve?

Step 1: Isolate the Trig Expression

Treat as a single quantity — call it

Replace :

Check: Isolate the Trig Expression

Isolate the trig expression in:

Do not solve — just isolate.

Check Answer: Trig Expression Now Isolated

Now ready: apply to both sides.

Step 2: Applying the Inverse Trig Function

From isolation:

Apply to both sides:

The arcsin gives the angle whose sine is 0.52.

Use a calculator: radians.

Calculator Check: Radian Mode Is Required

Always check: arcsin(0.52) should be ≈ 0.5464, NOT ≈ 31.3

If you get 31.3: switch to radian mode.

Step 3: Solve for the Original Variable

From:

Solve for :

The rider first reaches 40 feet at approximately minutes.

The Principal Value Is NOT the Full Answer

gives ONE angle in

This produces ONE value of min.

But: sine is periodic — there are more times when height = 40 ft.

We find those in Lesson 2.

A Second Example: Using arccos

Model: — temperature in °F after hours

Question: When is the temperature 57.6°F? Isolate and apply arccos.

Isolation:

Check: Solve for from arccos

Solve for .

Show the algebra step before advancing.

Check Answer: Hours

About 3 hours and 35 minutes after midnight — early morning.

Key Steps for the Principal Solution

1. Set up: substitute the target value
2. Isolate: algebra until trig expression
3. Inverse trig: apply , , or
4. Solve for : multiply or divide to finish

Use radian mode.

Inverse trig gives the angle — then isolate .

Coming Up in Lesson 2

Finding all solutions:

• Sine symmetry: second solution in period = minus first
• Adding full periods for more solutions
• Identifying all solutions in a given interval

Filtering and interpretation:

• Which solutions are physically meaningful?
• Stating answers with context and units