Lesson Summary

In this lesson we have examined relationships between and among the trig functions. The reciprocal identities tell us the relationship between pairs of trig functions that are reciprocals of each other. The quotient identities tell us relationships among functions in threes: the tangent function is the quotient of the sine and cosine functions, and the cotangent function is the reciprocal of this quotient. The Pythagorean identities, which rely on the Pythagorean theorem, also tell us relationships among functions in threes. Each identity can be used to find values of trig functions, and as well as to prove other identities, which will be a focus of chapter 3. We can also use identities to determine the domain and range of functions, which will be useful in chapter 2, where we will graph the six trig functions.

Points to Consider

How do you know if an equation is an identity? [hint: you could consider using a the calculator and graphing a related function, or you could try to prove it mathematically.]
How can you verify the domain or range of a function?

Review Questions

Use reciprocal identities to give the value of each expression.
1. $\sec\ \theta = 4, \cos \theta = ?$
2. $\sin \theta = \frac{1} {3}, \csc \theta = ?$
In the lesson, the range of the cosecant function was given as: or .
1. Use a calculator to fill in the table below. Round values to $4\;\mathrm{decimal}$ places.
2. Use the values in the table to explain in your own words what happens to the values of the cosecant function as the measure of the angle approaches $0\;\mathrm{degrees}$ .
3. Explain what this tells you about the range of the cosecant function.
4. Discuss how you might further explore values of the sine and cosecant to better understand the range of the cosecant function.

Angle	Sin	Csc
$10$
$5$
$1$
$0.5$
$0.1$
$0$
$-.1$
$-.5$
$-1$
$-5$
$-10$

3. In the lesson the domain of the secant function were given:

Domain: $\theta \epsilon^0, \theta \neq 90,270,450 \ldots$

Explain why certain values are excluded from the domain.

4. State the quadrant in which each angle lies, and state the sign of each expression

1. $\sin(80^\circ )$
2. $\cos(200^\circ )$
3. $\cot(325^\circ )$
4. $\tan(110^\circ )$

5. If $\cos \theta = \frac{6} {10}$ and $\sin \theta = \frac{8} {10}$ , what is the value of $\tan \theta$ ?

6. Use quotient identities to explain why the tangent and cotangent function have positive values for angles in the third quadrant.

7. If $\sin \theta = 0.4$ , what is the value of $\cos \theta$ ? Assume that $\theta$ is an angle in the first quadrant.

8. If $\cot \theta = 2$ , what is the value of $\csc \theta$ ? Assume that $\theta$ is an angle in the first quadrant.

9. Show that $1 + \tan^{2}\ \theta = \sec^{2}\ \theta$ .

10. Explain why it is necessary to state the quadrant in which the angle lies for problems such as #7.

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