5.4E: Describing Relationships with the Tangent Function (Exercises)

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Section 5.4 Exercises

Part 1: Solving equations with tangent

In exercises #1-4, find all the solutions to the equations.

1. $\tan \left(\theta \right)\ = 0$

2. $\tan \left(5 \theta \right)\ = 1 $

3. $3\tan \left(2 \theta \right)\ +1 = 0$

4. $4\tan \left(\dfrac{\theta}{4} \right)\ -3 = 0$

In exercises #5 - 6, find all solutions in the interval $ -2 \pi \le \theta \le 2 \pi $. Express your results in radians.

5. $\tan \left(3 \theta \right)\ = \sqrt{3}$

6. $\tan^{2} \left( \theta \right)\ - 1 = 0$

Part 2: Graphing with the tangent function

In exercises #7 - 12, find the period and horizontal shift of each of the following functions. Then graph the functions over the indicated intervals.

7. $y=\tan \left(\dfrac{1}{2} \theta \right)$ on $ -2 \pi \le \theta \le 2 \pi $

8. $y=2 \tan \left(3 \theta \right)$ on $ - \pi \le \theta \le \pi $

9. $y=2 \tan \left(\dfrac{1}{3} \theta \right)$ on $ -3 \pi \le \theta \le 3 \pi $

10. $y= \tan \left( \theta + \dfrac{\pi}{4}\right)$ on $ - \pi \le \theta \le \pi $

11. $y= \tan \left(\dfrac{\theta}{3} - \dfrac{\pi}{3}\right)$ on $ - \pi \le \theta \le \pi $

In exercises #12-13, find a possible formula for each function graphed below.

12. $A graph of a tangent function with a period change.$ 13. $A graph of a tangent function with a period change and phase shift.$

14. Identify the mistake(s) that was(were) made.

Graph $y=\tan \left(4 \theta \right)$

Step 1: Calculate the period $P=\dfrac{\pi}{B}= \dfrac{\pi}{4} $

Step 2: Find the x-intercept $4 \theta = \dfrac{\pi}{2} $

$ \theta = \dfrac{\pi}{8} $

Step 3: Find the vertical asymptototes $4 \theta = 0 $ and $4 \theta = \pi $

$ \theta = 0 $ and $ \theta = \dfrac{\pi}{4} $

Step 4: Then graph

$the graph of a tangent function$

Answer

2. $\theta = =9 ^{\circ} + 36 ^{\circ} n $

8. The period is $P=\dfrac{\pi}{3}$. There is no horizontal shift.

$The graph of a tangent function with a period change.$

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