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Trigonometry · Trigonometry: Similar Triangles, Right Triangles and Acute Triangles

Trigonometry: Similar Triangles, Right Triangles and Acute Triangles: Unit Review

1.1 1.2 1.3 2.1 2.2 2.3 3.1 3.2 3.3 3.4About 40 minutes
Learning goals
  • I can use similar triangles to solve problems, including shadows, mirrors, models and surveying.
  • I can solve right triangles with the primary trigonometric ratios and the Pythagorean theorem.
  • I can solve acute triangles with the sine law and the cosine law.
  • I can choose the right tool and communicate a clear solution.

Words to know

similar triangles
Triangles with equal corresponding angles and proportional corresponding sides.
primary trigonometric ratios
sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj.
angle of elevation / depression
The angle up / down from the horizontal to a line of sight.
sine law
a/sin A = b/sin B = c/sin C.
cosine law
a² = b² + c² − 2bc cos A.

Let's review!

In this unit you started with similar triangles: equal angles, proportional sides, and the AA, SSS and SAS conditions. You used them to measure trees, flagpoles and rivers. Then similar right triangles gave you the primary trigonometric ratios, sine, cosine and tangent, which you used to find sides and angles, angles of elevation and depression, and bearings. Finally you developed the sine law and the cosine law to solve acute triangles that have no right angle.

This review has a few questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Do one or two sections at a time. Draw a sketch for every question, even when one is not asked for. Keep your calculator in degree mode (sin 30° = 0.5). Round only final answers: lengths to one decimal place and angles to the nearest degree unless the question says otherwise. For every question you get wrong, write down why (wrong tool? wrong side labelled? rounding? arithmetic?) and go back to that lesson's worked examples. Finish with the 10-question quiz.

Similar trianglesAA, SSS or SAS similarityk = new ÷ originalcorresponding sides proportionalRight trianglessin θ = opp/hyp, cos θ = adj/hyptan θ = opp/adja² + b² = c²Sine law (acute)a/sin A = b/sin B = c/sin Cuse with a complete pair(AAS, ASA, SSA)Cosine law (acute)a² = b² + c² − 2bc cos Ause with SAS or SSScos A = (b² + c² − a²)/(2bc)
Your trigonometry toolbox. Similar triangles are the foundation; the trigonometric ratios work in right triangles; the sine and cosine laws work in acute triangles.
Just for fun

You now know enough trigonometry to measure a tree without climbing it, find the width of a river without getting wet, and locate a fire from 15 km away. Not bad for a few letters: S, O, H, C, A, H, T, O, A.

Congruent and Similar Figures

1

△ABC ≅ △DEF. Which side is equal to BC?

2

△PQR ~ △STU with PQ = 5, ST = 15 and QR = 7. Find TU.

3

Is a 4 cm by 10 cm rectangle similar to a 6 cm by 15 cm rectangle?

Investigating Similar Triangles: AA, SSS and SAS Similarity

1

One triangle has angles 48° and 67°. Another has angles 67° and 65°. Are they similar?

2

Are triangles with sides 3, 5, 6 and 6, 10, 12 similar?

3

In △ABC, DE ∥ BC with D on AB and E on AC. Why is △ADE ~ △ABC?

Finding Unknown Lengths in Similar Triangles

1

DE ∥ BC, AD = 4, AB = 10 and DE = 6. Find BC.

2

In a bow tie with AB ∥ DE (△ABC ~ △EDC), AC = 5, CE = 8 and AB = 6. Find DE.

3

DE ∥ BC, AD = x, DB = 5, DE = 3 and BC = 8. Find x.

Similar Triangles in the Real World: Shadows, Mirrors, Models and Surveying

1

A 1.5 m student casts a 2.5 m shadow. At the same time a building casts a 30 m shadow. How tall is the building?

2

Eyes 1.6 m high, 2 m from a mirror on the ground; the mirror is 11 m from a tree. How tall is the tree?

3

A 1:200 model of a building is 15 cm tall. How tall is the real building?

Side Ratios in Right Triangles: Sine, Cosine and Tangent

1

A right triangle has sides 9, 40 and 41. For the angle θ opposite 9, write sin θ, cos θ and tan θ.

2

Find tan 58° to four decimal places.

3

Why is sin 35° the same in every right triangle with a 35° angle?

Using Trigonometric Ratios to Find Side Lengths

1

Hypotenuse 14 cm, angle 33°. Find the side opposite the angle.

2

Adjacent side 9 m, angle 58°. Find the hypotenuse.

3

Opposite side 5.5 cm, angle 24°. Find the adjacent side.

Using Inverse Trigonometric Ratios to Find Angles

1

Opposite 7, hypotenuse 12. Find θ.

2

Opposite 11, adjacent 6. Find θ.

3

A ramp rises 0.6 m over a horizontal run of 7.5 m. Find its angle to one decimal place.

Solving Right Triangles

1

Solve △ABC with ∠C = 90°, a = 9 and c = 15.

2

Solve △ABC with ∠C = 90°, ∠B = 22° and c = 30 cm.

3

Why should you avoid using rounded values in later steps?

Angles of Elevation and Depression

1

From 40 m away, with eyes 1.5 m high, the angle of elevation to the top of a tower is 35°. How tall is the tower?

2

From the top of an 80 m cliff, the angle of depression to a boat is 15°. How far is the boat from the base of the cliff?

3

A 30 m tower is 50 m away on level ground. What is the angle of elevation to its top (ignore eye height)?

Multi-Step Problems: Two Right Triangles, Navigation and Surveying

1

A boat sails 12 km south, then 9 km west. Find its distance and bearing from the start.

2

From 60 m away, the angles of elevation to the bottom and top of a sign on a building are 28° and 35°. How tall is the sign?

3

From a 30 m lighthouse, the angles of depression to two boats in line are 25° and 15°. How far apart are the boats?

The Sine Law

1

In △ABC, ∠A = 48°, ∠B = 70° and a = 15 cm. Find b.

2

In acute △ABC, a = 12, b = 10 and ∠A = 72°. Find ∠B.

3

In which two situations can the sine law NOT start?

The Cosine Law

1

In △ABC, b = 8, c = 11 and ∠A = 55°. Find a.

2

A triangle has sides 7, 10 and 11. Find the angle opposite the side of length 11.

3

How is the cosine law related to the Pythagorean theorem?

Aa = ?Bb = ?Cc = 20 cm40°75°?Two angles + a side,but no complete pair yet.Find ∠C first, thenuse the sine law.
The first question in the next section, drawn to scale: two angles and the side between them.

Choosing the Right Tool: Solving Acute Triangles

1

Solve △ABC with ∠A = 40°, ∠B = 75° and c = 20 cm.

2

Which tool first? (a) two sides and the contained angle; (b) a right triangle with one side and one acute angle; (c) ∠A, ∠C and side b.

3

A triangle has sides 10, 12 and 14. Is it acute? Explain.

Solving Real-World Problems with Acute Triangles

1

Two lookout towers are 12 km apart. The angles from the line between them to a fire are 55° at tower A and 65° at tower B. How far is the fire from tower A?

2

Two boats leave a dock on straight courses 50° apart. One is 5 km away and the other 7 km away. How far apart are they?

3

What should a complete solution to a word problem include?

Check your understanding

Quiz

1.

Similar triangles always have:

2.

In △ABC, DE ∥ BC, AD = 3, AB = 9 and DE = 4. What is BC?

3.

A 2 m post casts a 1.6 m shadow. A tree casts a 12 m shadow at the same time. How tall is the tree?

4.

In a right triangle, tan θ equals:

5.

The hypotenuse is 10 cm and one angle is 40°. How long is the side opposite the 40° angle?

6.

tan θ = 1.2. What is θ to the nearest degree?

7.

Why is the angle of depression from a cliff to a boat equal to the angle of elevation from the boat to the cliff top?

8.

Which information lets you start with the sine law?

9.

In △ABC, b = 4, c = 5 and ∠A = 60°. What is a?

10.

The cosine law becomes the Pythagorean theorem when the angle is:

For parents and teachers

Use the review to find what to revisit:

  • Mixing up which sides match, or using part of a side in nested triangles → Lessons 1 to 3 (Congruent and Similar Figures; AA, SSS and SAS Similarity; Finding Unknown Lengths).
  • Shadow, mirror or scale-model problems set up wrong → Lesson 4 (Similar Triangles in the Real World).
  • Labelling opposite, adjacent and hypotenuse incorrectly, or choosing the wrong ratio → Lessons 5 and 6 (Sine, Cosine and Tangent; Finding Side Lengths).
  • Errors with sin⁻¹, cos⁻¹, tan⁻¹ (or calculator errors) → Lesson 7 (Inverse Ratios). Check that the calculator is in degree mode.
  • Incomplete solutions or rounding errors → Lesson 8 (Solving Right Triangles).
  • Angles measured from the wrong line, or forgetting eye height → Lesson 9 (Angles of Elevation and Depression).
  • Two-triangle problems and bearings → Lesson 10 (Multi-Step Problems).
  • Pairing a side with the wrong angle → Lesson 11 (The Sine Law).
  • Order-of-operations errors in a² = b² + c² − 2bc cos A → Lesson 12 (The Cosine Law).
  • Choosing the wrong law → Lesson 13 (Choosing the Right Tool).
  • Weak sketches, or not answering the question asked → Lesson 14 (Real-World Problems with Acute Triangles).

In Grade 11 (Functions, MCR3U) these ideas are extended to angles greater than 90°, obtuse triangles and the ambiguous case of the sine law.