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Integrated Practice: Mixed Trig Ratio and Inverse Trig Problems with Clear Workflows

Capítulo 8

Estimated reading time: 8 minutes

How to Use This Integrated Practice

This chapter is a cumulative workout: each set mixes skills and requires you to (1) identify the reference angle, (2) label sides relative to that angle, (3) choose an appropriate trig ratio, and (4) show a clean workflow. Every problem includes a required one-line justification for why you chose sine/cosine/tangent. Some problems reuse the same triangle but switch the reference angle to force careful side identification.

Workflow Template (Use for Every Problem)

Step 1: Mark the reference angle (write “ref = ___”).
Step 2: Label sides relative to that angle: opp, adj, hyp.
Step 3: Choose a ratio and write a one-line justification: “I use ___ because I know ___ and need ___.”
Step 4: Write the equation (ratio = number).
Step 5: Solve (algebra + calculator if needed).
Step 6: Reasonableness check (angle size vs side sizes; ratio between 0 and 1 for sine/cosine).

Set A — Label Sides and Compute Trig Ratios

In this set, you are not solving for unknowns. You are practicing correct side identification and ratio construction. Use exact simplified fractions and radicals when possible.

A1. Same triangle, different reference angle (forces careful labeling)

A right triangle has legs 6 and 8, and hypotenuse 10. The right angle is between the legs. Let angle A be opposite the side of length 6, and angle B be opposite the side of length 8.

Task: Compute sin A, cos A, tan A.
Task: Compute sin B, cos B, tan B.
Required one-line justification (example format): “For tan A I use tangent because I’m comparing opposite and adjacent relative to angle A.”

Ratio	Angle A	Angle B
`sin`	___	___
`cos`	___	___
`tan`	___	___

A2. Identify the “adjacent” correctly (not always the longer leg)

A right triangle has hypotenuse 13, one leg 5, and the other leg 12. Angle θ is adjacent to the leg of length 5.

Compute sin θ, cos θ, tan θ.
Write one sentence explaining why the hypotenuse is the denominator in sine and cosine.

A3. Ratio from a diagram description (no picture given)

In a right triangle, angle θ has opposite side 9 and adjacent side 40.

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Compute tan θ.
Compute sin θ and cos θ by first finding the hypotenuse.
Justification line: Choose one ratio and justify it in one line (your choice).

Set B — Solve for a Missing Side (Trig Ratio Problems)

For each problem: label sides relative to the given angle, choose a ratio, justify the choice in one line, then solve. Round lengths to the nearest tenth unless stated otherwise.

B1. Find a leg using sine

A right triangle has hypotenuse 18 and an acute angle of 35°. Find the length of the side opposite the 35° angle.

Justification line: “I use ___ because I know ___ and need ___.”
Equation line: Write the trig equation before solving.

B2. Find a leg using cosine

A right triangle has hypotenuse 22 and an acute angle of 61°. Find the length of the side adjacent to the 61° angle.

Include your one-line ratio justification.

B3. Find a leg using tangent (no hypotenuse given)

A right triangle has an acute angle of 28°. The side adjacent to the 28° angle is 14. Find the opposite side.

Include your one-line ratio justification.
After solving, state whether the opposite should be smaller or larger than 14 and why (based on the angle).

B4. Same triangle, different reference angle (forces re-labeling)

In a right triangle, the legs are 9 and 12. Let angle A be adjacent to the leg of length 12, and angle B be adjacent to the leg of length 9.

Given A = 36° (treat this as measured), find the hypotenuse using cosine relative to angle A.
Then, using the same triangle but reference angle B, find the hypotenuse again using sine relative to angle B (you will need B = 90° − A).
Justification line: Write one line for each method explaining why cosine fits A and why sine fits B.

Note: This problem is about consistent side labeling. The hypotenuse is the same physical side both times, but “opposite” and “adjacent” switch when the reference angle switches.

Set C — Solve for a Missing Angle (Inverse Trig in Context)

For each problem: identify which sides are known relative to the unknown angle, choose the correct inverse trig function, justify the choice in one line, then compute the angle (nearest tenth of a degree).

C1. Angle from opposite and hypotenuse

In a right triangle, the side opposite angle θ is 7 and the hypotenuse is 20. Find θ.

Justification line: “I use inverse ___ because I know ___ and ___.”
Write the equation in the form sin θ = ... (or your chosen ratio) before applying the inverse.

C2. Angle from adjacent and hypotenuse

In a right triangle, the side adjacent to angle θ is 15 and the hypotenuse is 17. Find θ.

Include your one-line ratio justification.

C3. Angle from opposite and adjacent

In a right triangle, the side opposite angle θ is 24 and the side adjacent is 10. Find θ.

Include your one-line ratio justification.
After computing θ, state whether it should be closer to 0°, 45°, or 90° and why (based on the ratio size).

C4. Same triangle, different reference angle (angles swap roles)

A right triangle has legs 8 and 15 and hypotenuse 17. Let angle A be opposite the side of length 8, and angle B be opposite the side of length 15.

Find A using an inverse trig function of your choice (but justify it).
Find B using a different inverse trig function than you used for A (and justify it).
Verify that A + B = 90° (allow small rounding error).

Set D — Word Problems: Interpret, Diagram, Solve

For each word problem: (1) sketch a right-triangle diagram, (2) label the angle and sides with words (height, shadow, horizontal distance, line of sight), (3) choose a trig ratio with a one-line justification, (4) solve with units.

D1. Ladder against a wall (choose the correct ratio)

A 16 ft ladder leans against a vertical wall. The angle between the ladder and the ground is 68°. How high up the wall does the ladder reach?

Diagram labels: ladder = hypotenuse, height = opposite (relative to the ground angle), ground distance = adjacent.
Justification line: Explain in one line why sine or cosine is the best choice here.

D2. Ramp design (tangent vs sine/cosine decision)

A wheelchair ramp rises 2.5 ft vertically over a horizontal run. The ramp makes a 6° angle with the ground. Find the horizontal run needed (in feet).

Justification line: Explain in one line why tangent is a natural first choice (or explain why you chose a different ratio).
State whether your run should be much larger than 2.5 ft and why (based on the small angle).

D3. Angle of elevation with a measured distance (careful: which distance is given?)

A person stands 45 m from the base of a building (horizontal distance). The angle of elevation to the top is 32°. Find the building’s height (ignore the person’s height).

Justification line: Explain in one line why tangent fits the given information.
Include units in your final answer.

D4. Sloped cable (hypotenuse is the cable, not the ground)

A support cable runs from the top of a 9 m pole to a point on the ground 12 m from the base of the pole. Find (1) the cable length and (2) the angle the cable makes with the ground.

Part (1): You may use the Pythagorean theorem or trig; choose one and state why.
Part (2) justification line: Explain in one line why inverse tangent is appropriate for the angle with the ground.

D5. Same physical triangle, different reference angle (interpretation matters)

A drone is flying so that the line of sight from an observer to the drone is 120 m. The drone is 72 m above the ground. Assume the observer is on level ground directly away from the point below the drone.

Part (a): Find the angle of elevation from the observer to the drone.
Part (b): Find the angle between the line of sight and the vertical (at the drone).
Justification lines: One line for (a) and one line for (b), explaining your trig choice and how the reference angle changes which side is “opposite.”

Short Self-Check: Verify Using an Alternate Method

Use these to confirm your own solutions. The goal is to verify with a different ratio or a different viewpoint, not to redo the same steps.

Self-Check 1 (side verification via a second ratio)

From problem B3 (adjacent 14, angle 28°), you found the opposite side o using tangent.

Check: Compute the hypotenuse two ways and compare:
- Method 1: Use sin 28° = o/h to solve for h.
- Method 2: Use cos 28° = 14/h to solve for h.
State whether the two values of h match within rounding.

Self-Check 2 (angle verification using a complementary angle)

From problem C4, you computed angles A and B in the 8–15–17 triangle.

Check: Compute A using a different inverse function than before (for example, if you used sin⁻¹(8/17), now use tan⁻¹(8/15)), and compare.
Then compute B as 90° − A and compare to your earlier B.

Self-Check 3 (word-problem consistency check)

From problem D1 (ladder), after finding the height h, verify by finding the ground distance d using cosine, then confirm that h^2 + d^2 is approximately 16^2 (allow rounding error).

1) d = 16 cos(68°)  2) Check: h^2 + d^2 ≈ 256

Now answer the exercise about the content:

In a right triangle, angle θ has opposite side 24 and adjacent side 10. Which setup correctly finds θ, and what should θ be closest to?

You are right! Congratulations, now go to the next page

You missed! Try again.

With opposite and adjacent known, tangent matches: tan θ = opp/adj = 24/10, so θ = tan⁻¹(24/10). A ratio greater than 1 suggests an angle larger than 45°, closer to 90°.