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Virginia SOL Mathematics Textbook

Geometry Workbook — Chapter 9: Right Triangle Trigonometry

SOL G.TR.4 (d, e, g) · Companion to Textbook Chapter 9

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 112.

PAGE 1 — Chapter opener

Chapter 9 · Right Triangle Trigonometry

Standard G.TR.4 (d, e, g)

In this chapter you will:

Words to know: opposite · adjacent · hypotenuse · sine · cosine · tangent · SOH-CAH-TOA · ratio · inverse function · complementary · angle of elevation · angle of depression · line of sight · horizontal

Conventions: name the angle first. A ratio is unitless. Sides to the nearest hundredth, angles to the nearest degree, both from unrounded values. Degrees, not radians.

PAGE 2 — Naming the sides

9.1 Opposite, Adjacent, Hypotenuse

FIGURE: fig1-opposite-adjacent-hypotenuse.png (full width)

Fill in the blanks.

The hypotenuse is opposite the ____________ angle and never ____________ .

Opposite and adjacent ____________ when you change which acute angle you work from.

  1. Which side is the hypotenuse? ______ Does it change between panels? ______

  2. Relative to ∠A\angle A: the 33 is ____________ , the 44 is ____________ .

  3. Relative to ∠B\angle B: the 33 is ____________ , the 44 is ____________ .

PAGE 3 — The three ratios

SOH CAH TOA

FIGURE: fig2-soh-cah-toa.png (full width)

  1. Give the three definitions.

    sin⁡θ=\sin \theta = ____________ cos⁡θ=\cos \theta = ____________ tan⁡θ=\tan \theta = ____________

PAGE 4 — Six ratios, one triangle

Read Them Off

FIGURE: fig3-ratios-from-a-3-4-5.png (full width)

  1. sin⁡A=\sin A = ______ cos⁡A=\cos A = ______ tan⁡A=\tan A = ______

  2. sin⁡B=\sin B = ______ cos⁡B=\cos B = ______ tan⁡B=\tan B = ______

PAGE 5 — Practice · finding ratios

Practice

Legs 88 and 1515, hypotenuse 1717; ∠A\angle A opposite the 88, ∠B\angle B opposite the 1515.

  1. sin⁡A=\sin A = ______

  2. cos⁡A=\cos A = ______

  3. tan⁡A=\tan A = ______

  4. sin⁡B=\sin B = ______

  5. cos⁡B=\cos B = ______

  6. tan⁡B=\tan B = ______

  7. Legs 55 and 1212. Hypotenuse ______ tangent of the angle opposite the 55 = ______

  8. sin⁡θ=725\sin \theta = \tfrac{7}{25}. Adjacent = ______ cos⁡θ=\cos \theta = ______ tan⁡θ=\tan \theta = ______

  9. tan⁡θ=34\tan \theta = \tfrac{3}{4}. Hypotenuse = ______ sin⁡θ=\sin \theta = ______ cos⁡θ=\cos \theta = ______

Legs 99 and 4040, hypotenuse 4141; ∠A\angle A opposite the 99.

  1. sin⁡A=\sin A = ______

  2. cos⁡A=\cos A = ______

  3. tan⁡A=\tan A = ______

PAGE 6 — Apply and reason · Lesson 9.1

Think It Through

  1. Why does a trigonometric ratio have no units?

  2. Error analysis. A student writes sin⁡A=adjacenthypotenuse\sin A = \tfrac{\text{adjacent}}{\text{hypotenuse}}. What did they write instead, and what is correct?

  3. Reasoning. Why are there exactly three ratios?

  4. State the three definitions in words. ____________________

PAGE 7 — Why a ratio belongs to the angle

9.2 The Reason Trigonometry Works

FIGURE: fig4-similar-triangles-same-ratio.png (full width)

Fill in the blanks.

The three triangles are similar by ______ , so their sides are ____________ and every ratio is the ______ .

  1. What do the three triangles have in common? ____________________

  2. Criterion: ______ Why that forces one tangent: ____________________

  3. tan⁡θ=\tan \theta = ______

PAGE 8 — Verifying against a calculator

Find It, Then Check It

FIGURE: fig5-verify-against-the-calculator.png (full width)

  1. Name the two routes to the same number. ____________________

  2. What does a disagreement usually mean? ____________________

PAGE 9 — Verifying against the complement

Sine and Cosine Trade Places

FIGURE: fig6-sine-and-cosine-swap.png (full width)

Fill in the blanks.

sin⁡θ=cos⁡(\sin \theta = \cos( ______ )) and cos⁡θ=sin⁡(\cos \theta = \sin( ______ ))

Also, for any acute angle, sin⁡2θ+cos⁡2θ=\sin^2 \theta + \cos^2 \theta = ______ — which is the ____________________ in ratio form.

  1. sin⁡37°=\sin 37° = ______ cos⁡53°=\cos 53° = ______

PAGE 10 — Practice · verifying

Practice

  1. 66-88-1010, ∠A\angle A opposite the 66. Verify sin⁡A=35\sin A = \tfrac{3}{5}: ____________

  2. 99-1212-1515, ∠A\angle A opposite the 99. Verify tan⁡A=34\tan A = \tfrac{3}{4}: ____________

  3. sin⁡30°=0.5\sin 30° = 0.5. cos⁡60°=\cos 60° = ______

  4. cos⁡20°≈0.9397\cos 20° \approx 0.9397. sin⁡70°=\sin 70° = ______

  5. sin⁡45°=cos⁡\sin 45° = \cos ______ °°

  6. sin⁡θ=cos⁡θ\sin \theta = \cos \theta when θ=\theta = ______

  7. 55-1212-1313: sin⁡2A+cos⁡2A=\sin^2 A + \cos^2 A = ______ + ______ = ______

  8. 33-44-55: sin⁡Acos⁡A=\dfrac{\sin A}{\cos A} = ______ = tan⁡A\tan A ✓

  9. One report: sin⁡θ=0.6\sin \theta = 0.6. Another: θ=37°\theta = 37°. Agree? ______ Check: ____________

  10. sin⁡25°≈0.4226\sin 25° \approx 0.4226. cos⁡65°=\cos 65° = ______

  11. 1616-3030-3434, ∠A\angle A opposite the 1616. Verify tan⁡A=815\tan A = \tfrac{8}{15}: ____________

  12. cos⁡θ=0.8\cos \theta = 0.8, θ\theta acute. sin⁡θ=\sin \theta = ______

PAGE 11 — Apply and reason · Lesson 9.2

Think It Through

  1. Application. A ramp rises 33 ft over a 1212 ft run. tan⁡θ=\tan \theta = ______ θ=\theta = ______ °

  2. Error analysis. A student writes sin⁡50°+cos⁡50°=1\sin 50° + \cos 50° = 1. Check it, and state the identity intended:

  3. Reasoning. Why is sin⁡θ=cos⁡(90°−θ)\sin \theta = \cos(90° - \theta)?

  4. Name two ways to verify a ratio. ____________________

PAGE 12 — Choosing the ratio

9.3 Pick the One That Mentions Both

FIGURE: fig7-choose-the-ratio.png (full width)

The three steps.

  1. ____________ the sides relative to the given angle.

  2. ____________ the ratio that mentions both.

  3. ____________ , rounding once at the end.

  4. Hypotenuse given, opposite wanted → ______

  5. Adjacent given, hypotenuse wanted → ______

PAGE 13 — Where the unknown lands

Multiply or Divide

FIGURE: fig8-solve-for-a-side.png (full width)

  1. Equation for xx: ____________________

  2. x=x = ______

  3. Equation for hh: ____________________

  4. h=h = ______

PAGE 14 — Practice · missing sides

Practice

  1. 28°28°, hypotenuse 2020 → opposite ______

  2. 28°28°, hypotenuse 2020 → adjacent ______

  3. 52°52°, adjacent 1414 → opposite ______

  4. 52°52°, adjacent 1414 → hypotenuse ______

  5. 63°63°, opposite 3030 → hypotenuse ______

  6. 63°63°, opposite 3030 → adjacent ______

  7. 15°15°, hypotenuse 88 → opposite ______ adjacent ______

  8. 71°71°, adjacent 55 → opposite ______

  9. 33°33°, hypotenuse 1515 → opposite ______

  10. 33°33°, adjacent 1515 → opposite ______

  11. 33°33°, opposite 1515 → hypotenuse ______

PAGE 15 — Apply and reason · Lesson 9.3

Think It Through

  1. Application. A 2424 ft ramp at 11°11°. Rise = ______ ft

  2. Application. A guy wire from a 4040 ft pole, meeting the ground at 55°55°. Length = ______ ft

  3. Error analysis. Both known sides are legs and a student uses sine. Why it cannot work, and the right ratio:

  4. Reasoning. Why does the unknown's position decide multiply versus divide?

  5. Application. A kite on 120120 ft of string at 48°48°. Height = ______ ft

  6. Error analysis. A student rounds sin⁡35°\sin 35° to 0.570.57 first. Their answer ______ Correct ______ Why: ____________

  7. State the three steps for a missing side. ____________________

PAGE 16 — Running it backwards

9.4 From Two Sides to the Angle

FIGURE: fig9-solve-for-an-angle.png (full width)

Fill in the blanks.

An inverse function takes a ______ back to the ______ that has it.

A calculator in ____________ mode returns 0.64350.6435 where 36.87°36.87° was wanted.

  1. Which two sides are given? ______ and ______

  2. Which ratio names both? ______

  3. Equation: ____________________

  4. θ=\theta = ______ °

  5. Unrounded: ______ °

  6. What does an inverse trigonometric function do? ____________________

PAGE 17 — Practice · missing angles

Practice

  1. Opposite 99, hypotenuse 1515 → θ=\theta = ______ °

  2. Adjacent 88, hypotenuse 1717 → θ=\theta = ______ °

  3. Opposite 55, adjacent 99 → θ=\theta = ______ °

  4. Opposite 1111, hypotenuse 1414 → θ=\theta = ______ °

  5. Opposite 2121, adjacent 2020 → θ=\theta = ______ °

  6. Legs 77 and 2424 → ______ ° and ______ ° Check: ______ + ______ = ______

  7. Opposite 66, hypotenuse 1010 → ______ °

  8. Adjacent 66, opposite 66 → ______ °

  9. Adjacent 1212, hypotenuse 1313 → ______ °

PAGE 18 — Apply and reason · Lesson 9.4

Think It Through

  1. Application. A ramp rises 22 ft over an 1818 ft run. Angle = ______ °

  2. Application. A 2525 ft ladder reaching 2323 ft up a wall. Angle = ______ °

  3. Error analysis. Given two legs, a student uses sin⁡−1\sin^{-1}. Why it is wrong, and the right function:

  4. Reasoning. Why do the two acute angles found this way always add to 90°90°?

  5. Application. A road rises 4040 m over 500500 m. Angle = ______ °

  6. Error analysis. A student reports 0.64°0.64°. Cause and correct answer:

  7. State the three steps for a missing angle. ____________________

PAGE 19 — Two names, one angle

9.5 Elevation and Depression

FIGURE: fig10-elevation-and-depression.png (full width)

Fill in the blanks.

Both angles are measured from a ____________ .

They are congruent because they are ____________________ angles across two ____________ lines.

  1. From what is each angle measured? ____________________

  2. Why are they congruent? Reason: ____________________

  3. Why is the angle of depression not measured from the cliff face?

PAGE 20 — An elevation problem

Looking Up

FIGURE: fig11-elevation-in-context.png (full width)

  1. Equation: ____________________ h=h = ______ ft

PAGE 21 — A depression problem

Looking Down

FIGURE: fig12-depression-in-context.png (full width)

  1. Which angle inside the triangle equals 24°24°? ______ Why: ____________

  2. Equation: ____________________ d=d = ______ ft

PAGE 22 — Practice · elevation and depression

Practice

  1. 8080 ft away, elevation 55°55° → height ______ ft

  2. 120120 ft away, elevation 38°38° → height ______ ft

  3. 6060 ft tall, elevation 41°41° → distance ______ ft

  4. 150150 ft lighthouse, depression 12°12° → distance ______ ft

  5. 200200 ft cliff, depression 35°35° → distance ______ ft

  6. 30,00030{,}000 ft, depression 7°7° → ground distance ______ ft ≈ ______ miles

  7. 200200 ft of string, elevation 55°55° → height ______ ft

  8. 100100 ft away, elevation 40°40° → height ______ ft

  9. 9090 ft tower, depression 20°20° → distance ______ ft

  10. 5050 ft tower, elevation 30°30° → distance ______ ft

PAGE 23 — Apply and reason · Lesson 9.5

Apply It

  1. Application. 4545 ft from a tree, eyes 55 ft up, elevation 62°62°.

    Above eye level = ______ ft Tree height = ______ ft

  2. Application. A ramp rising 33 ft at no more than 5°5°. Shortest run = ______ ft

  3. Application. From a boat, elevation 9°9° to a lighthouse top 120120 ft above the water. Distance = ______ ft

  4. Error analysis. A student places the 24°24° between the cliff face and the line of sight. Error and correct placement:

  5. Reasoning. Why do an elevation problem and its matching depression problem have the same answer?

  6. What is an angle of elevation measured from? ____________ An angle of depression? ____________

PAGE 24 — Work frames

Blank Frames

FIGURE: fig13-blank-trig-frames.png (full width)

Mark the right angle. Mark the given acute angle. Then label opposite, adjacent, and hypotenuse relative to it — and only then choose a ratio.

Reminder. Keep the ratio unrounded until the very last step. Check that the calculator is in degree mode before every angle answer.

PAGE 25 — Chapter review

Review

Review 1 (G.TR.4d). Legs 2020 and 2121, hypotenuse 2929; ∠A\angle A opposite the 2020.

Review 2 (G.TR.4e). A right triangle with a 37°37° angle and a hypotenuse of 2626.

Review 3 (G.TR.4g). A drone above a landing pad; from 140140 ft away on level ground the elevation is 47°47°.

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