Task Sets: Solve trigonometric equations - Common Core Math - HS Functions

A

Warm-Up: Review What You Know

1.

The function $\arcsin$ has a restricted domain. What is the range of $\arcsin$ ?

A.

$[0, \pi]$

B.

$[-\frac{\pi}{2}, \frac{\pi}{2}]$

C.

$[0, 2\pi]$

D.

$[-\pi, \pi]$

2.

A model gives $h(t) = 12\sin(\pi t / 6) + 15$ . Set $h(t) = 21$ . After isolating the sine, what value does $\sin(\pi t/6)$ equal?

Enter as a fraction (e.g., 1/2).

3.

A student evaluates $\arcsin(0.6)$ on a calculator and gets $36.87$ . What is the likely error?

A.

No error — $\arcsin(0.6) = 36.87$ in radians.

B.

The student should have used $\arccos$ instead of $\arcsin$ .

C.

The calculator is in degree mode. In radians, $\arcsin(0.6) \approx 0.6435$ .

D.

$\arcsin(0.6)$ is undefined because 0.6 is not a special angle value.

B

Fluency Practice

1.

A tide model is $H(t) = 3\cos(\pi t/6) + 5$ , where $t$ is hours. Set $H = 7$ .

After isolating the cosine, what value does $\cos(\pi t/6)$ equal?

Enter as a fraction (e.g., 2/3).

2.

A Ferris wheel model gives $h(t) = 20\sin(\pi t/4) + 25$ . You want to find when $h = 40$ .

After isolating the sine expression: $\sin(\pi t/4) = ?$

Enter as a decimal rounded to two decimal places (e.g., 0.75).

3.

Given $\sin(\theta) = 0.75$ (from the Ferris wheel problem above), find the principal value $\theta = \arcsin(0.75)$ .

Use a calculator (radian mode). Round to four decimal places.

4.

You found $\sin(\pi t/4) = 0.75$ , and the principal solution gives $\pi t/4 \approx 0.8481$ .

Within one period of length 8 (since the period is $2\pi / (\pi/4) = 8$ ), there is a second value of $\pi t/4$ that also gives $\sin = 0.75$ . What is it?

A.

$\pi - 0.8481 \approx 2.2935$ (supplementary angle, since sine is symmetric about $\pi/2$ )

B.

$2\pi - 0.8481 \approx 5.4351$ (wrong symmetry — this applies to cosine, not sine)

C.

$0.8481 + \pi \approx 3.9897$ (this would give a negative sine value)

D.

There is only one value — the calculator gives the complete answer.

5.

From the Ferris wheel problem: $\pi t/4 = 0.8481$ gives $t_1$ , and $\pi t/4 = 2.2935$ gives $t_2$ .

What is $t_1$ (the first time the Ferris wheel reaches height 40)? Round to two decimal places.

C

Mixed Practice

1.

A temperature model is $T(t) = 8\cos(\pi t/12) + 62$ , where $t$ is hours after midnight.

Find the principal solution (first positive time) when $T = 68$ . Give $t$ rounded to two decimal places.

Note: $\arccos(0.75) \approx 0.7227$ radians.

2.

Using the temperature model $T(t) = 8\cos(\pi t/12) + 62$ and the fact that $\arccos(0.75) \approx 0.7227$ , find the second solution (within the first period of length 24) for $T = 68$ .

Round to two decimal places.

3.

Using the graph above, which statement correctly describes the two solutions $t_1 \approx 2.76$ and $t_2 \approx 21.24$ for $T = 68$ ?

A.

Both solutions occur when the temperature is falling.

B.

$t_1$ occurs as the temperature is falling (early morning) and $t_2$ also occurs as the temperature is falling (late night). Both are on the descending branch of the curve.

C.

$t_1$ occurs as the temperature is falling from the nighttime high (near midnight) and $t_2$ occurs as the temperature is rising toward the next cycle's high.

D.

The two solutions are identical — there is only one time when $T = 68$ .

4.

The model $T(t) = 8\cos(\pi t/12) + 62$ (period = 24 hours). After finding solutions $t_1 \approx 2.76$ and $t_2 \approx 21.24$ in the first cycle, list all solutions in the interval $[0, 48]$ .

Enter the number of solutions in the interval $[0, 48]$ .

5.

A model gives solutions $t = -2.5$ and $t = 5.3$ hours. The problem states that $t = 0$ represents 6 AM and $t$ ranges from 0 to 24. Which solution is valid in context and why?

D

Word Problems

1.

A Ferris wheel is modeled by $h(t) = 25\sin(\pi t/4) + 27$ , where $h$ is height in feet and $t$ is minutes. The ride lasts 16 minutes (two full periods).

1.

Set $h = 40$ and isolate the sine. What does $\sin(\pi t/4)$ equal?

Enter as a decimal (e.g., 0.52).

2.

Given $\sin(\pi t/4) = 0.52$ , find the principal solution $t_1$ (when the wheel first reaches 40 feet).

Use $\arcsin(0.52) \approx 0.5464$ radians. Round $t_1$ to two decimal places.

3.

Find $t_2$ , the second time the Ferris wheel reaches 40 feet (within the first 8-minute period).

Use the supplementary angle: $\pi - 0.5464 \approx 2.5952$ .

2.

A tide model is $H(t) = 3.2\cos(\pi t/6.25) + 4.0$ , where $H$ is height in meters and $t$ is hours after midnight. The period is 12.5 hours.

1.

How many times does the tide reach exactly 6 meters during the first 24 hours? (Period = 12.5 hours, so there are almost 2 full cycles.)

Use the fact that each cycle gives 2 solutions.

2.

Daylight is from 6 AM to 6 PM (hours 6 to 18 after midnight). How would you determine which of the four tide solutions occur during daylight hours?

3.

A student solving a modeling problem finds $t = -1.5$ and $t = 10.5$ hours as solutions to $T(t) = 70$ , where $t = 0$ represents midnight. The problem asks for times between midnight and noon (0 to 12 hours).

The student discards $t = -1.5$ as invalid. Explain whether this is the right decision and why, paying careful attention to what $t = -1.5$ means in context.

E

Error Analysis

1.

A student solved: "Find all $t$ in $[0, 8]$ where $\sin(\pi t/4) = 0.52$ ."

Student work: $t = 4 \times \arcsin(0.52)/\pi \approx 4 \times 0.5464/\pi \approx 0.70$ . Answer: only $t \approx 0.70$ .

What did the student miss?

A.

The student computed $t_1$ correctly but forgot to find the second solution per period (from the supplementary angle) and any solutions in the second period.

B.

The student made an arithmetic error in computing $t_1$ .

C.

The student should have used $\arccos(0.52)$ instead of $\arcsin(0.52)$ .

D.

The student forgot to convert from degrees to radians.

2.

A student solved: $h(t) = 25\sin(\pi t/4) + 27 = 40$

Step 1: $\sin(\pi t/4) = 0.52$
Step 2: $t = \arcsin(0.52) \approx 0.5464$

The student reported $t \approx 0.5464$ minutes as the answer.

What error did the student make in Step 2?

A.

The student correctly identified the angle but forgot to solve for $t$ .

B.

The student used the wrong value — $\arcsin(0.52) \approx 31.3\degree$ , not $0.5464$ .

C.

The student should have found $t = \pi/4 \times \arcsin(0.52)$ .

D.

The student isolated the sine incorrectly in Step 1.

F

Challenge

1.

A water wheel model is $h(t) = 4\sin(\pi t/3) + 5$ , where $h$ is height in feet and $t$ is minutes. The wheel runs for 12 minutes.

How many times total does the wheel reach exactly 8 feet during the 12-minute run?

(Period = 6 minutes. Use $\arcsin(0.75) \approx 0.8481$ radians.)

2.