When simplifying, a possible strategy is to convert everything to sines and cosines and to divide out common factors where possible.
In Try This 3 you may have noticed that identities can be used to simplify expressions. A possible solution for Try This 3 is shown in Simplifying a Trigonometric Expression.
If you would like to see another example of using trigonometric identities to simplify expressions, read “Example 2” on pages 293 and 294 of the textbook. If you don't think you need to see another example, move ahead to Self-Check 2.
Self-Check 2
- Complete “Your Turn” from “Example 2” on page 294 of the textbook. Answer
- Complete questions 3, 4.a., and 4.b. on page 296 of the textbook. Answer
So far you have seen how to verify an identity and how to use identities to simplify expressions. In the next section, you will attempt to predict an identity beginning with a geometric representation.
Try This 4
From Module 4, you know that a point where the terminal arm of angle θ intersects the unit circle can be represented by (cos θ, sin θ), as shown in the diagram.
- Sketch the diagram shown and label the length of each side of the triangle.
- Use the Pythagorean theorem, a2 + b2 = c2, to predict a relationship between sine and cosine.
- Verify your prediction numerically and graphically.
- The identities 1 + tan2 θ = sec2 θ and 1 + cot2 θ = csc2 θ can also be predicted from a geometric representation similar to the one shown.
- Verify 1 + cot2 θ = csc2 θ numerically and graphically.
- Multiply each term in 1 + tan2 θ = sec2 θ by cos2 θ and simplify. What do you notice?
Save your answers in your course folder.
The lengths of the triangle are sin θ, cos θ, and 1.
