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C. Trigonometric Functions · Trigonometry of Acute Triangles

Trigonometry of Acute Triangles: Unit Review

C1.1 C1.2 C1.3 C1.4 C1.5About 40 minutes
Learning goals
  • I can solve right triangles and two-triangle problems with the primary ratios.
  • I can explain where the sine law and cosine law come from and verify them.
  • I can choose and use the sine law or the cosine law to solve acute triangles.
  • I can solve real-world problems in surveying, navigation and construction.

Words to know

primary trigonometric ratios
sin = opp/hyp, cos = adj/hyp, tan = opp/adj (right triangles only).
sine law
a/sin A = b/sin B = c/sin C.
cosine law
a² = b² + c² − 2bc cos A; for angles, cos A = (b² + c² − a²) ÷ (2bc).
bearing
A direction measured clockwise from north (true bearing) or as N/S, angle, E/W (compass bearing).

Let's review!

You started this unit with right triangles: naming sides, using sine, cosine and tangent to find sides and angles, and applying them to angles of elevation and depression and to bearings. Then you solved problems with two right triangles sharing a side, and stretched the same ideas into 3-D. Next you discovered and verified the sine law and the cosine law, learned when to use each one, and used them to solve real-world problems in surveying, navigation and construction.

Each section below matches one lesson. Answers are hidden; try each question first.

How to use this review

Do two or three sections at a time. Draw a sketch for every question, even when the question doesn't ask for one. 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. If you miss more than one question in a section, go back to that lesson (see the parent notes for which lesson matches which mistake).

Right triangleSOH CAH TOAPythagorasangles add to 90°Sine lawa/sin A = b/sin Bneeds a complete pairAAS, ASA, SSACosine lawa² = b² + c² − 2bc cos Ano pair neededSAS, SSSMany problems chain these together: solve one triangle, pass a side to the next.
Your toolbox for this unit. The question "Is there a complete opposite pair?" decides between the two laws.
Just for fun

You now know enough trigonometry to measure a cliff without climbing it, find the width of a river without getting wet, and locate a fire from two towers 15 km apart. Not bad for three letters: sin, cos and tan.

Right-Triangle Trigonometry: Finding Side Lengths

1

A 4.5 m ladder makes a 72° angle with the ground. How high up the wall does it reach?

2

In right △ABC (∠C = 90°), ∠A = 25° and BC = 7.0 cm. Find AB.

3

A tree casts a 9.0 m shadow when the sun is 50° above the horizon. How tall is the tree?

Finding Angles and Solving Right Triangles

1

A ramp rises 0.6 m over a horizontal run of 7.2 m. Find its angle.

2

Solve △PQR: ∠Q = 90°, p = 12 m, r = 9 m.

3

What does cos⁻¹(0.8) mean, and what is it?

Angles of Elevation and Depression, and Bearings

1

From a 30 m bridge, the angle of depression to a canoe is 22°. How far is the canoe horizontally from the point below the observer?

2

Write S 35° E as a true bearing.

3

A plane flies 300 km on a bearing of 120°. How far south and east of its start is it?

Two Right Triangles: Heights and Distances

1

Two people on opposite sides of a 30 m tower see its top at 35° and 48°. How far apart are they?

2

From a window 12 m up, the angle of elevation to the top of a building across the street is 36° and the angle of depression to its base is 27°. How tall is the building?

3

From A, a tower's top is at 30° elevation. From B, 25 m closer, it is at 45°. Find the height.

Two Right Triangles: Shared Sides, Navigation and Surveying

1

In △ABC, ∠B = 50°, ∠C = 65° and the altitude from A is 10 cm. Find BC.

2

A boat sails 8 km on 020°, then 6 km on 110°. Find its distance and bearing from the start.

3

A and B are 50 m apart on a riverbank. A tree across the river makes ∠A = 55° and ∠B = 60°. Find the river's width.

Right Triangles in Three Dimensions

1

Find the space diagonal of a 3 m by 4 m by 12 m box.

2

For a 40 cm by 30 cm by 20 cm box, find the angle between the space diagonal and the 40 by 30 base.

3

A 25 m pole is seen from a point due south of it at 42° elevation and from a point due east at 35°. How far apart are the points?

Discovering and Verifying the Sine Law

1

Measured: a = 7.4 cm with ∠A = 52°, b = 8.9 cm with ∠B = 71°. Do these support the sine law?

2

In △ABC, ∠A = 44°, ∠B = 76°, a = 10 m. Find b.

3

Explain how the altitude from C shows that b sin A = a sin B.

Using the Sine Law to Find Angles and Solve Triangles

1

In acute △ABC, a = 16, b = 13, ∠A = 68°. Find ∠B.

2

Solve acute △XYZ: ∠X = 62°, x = 20, y = 17.

3

A calculation gives sin B = 1.3. What does it mean?

Discovering and Verifying the Cosine Law

1

In △ABC, b = 8 cm, c = 12 cm and ∠A = 61°. Find a.

2

What does the cosine law become when ∠A = 90°?

3

Write a spreadsheet formula for a when b is in B2, c in C2 and ∠A (degrees) in D2.

Using the Cosine Law to Find Angles and Solve Triangles

1

Find the largest angle of a triangle with sides 9, 10 and 12.

2

Solve the triangle with sides 7, 8 and 9.

3

Is a triangle with sides 6, 8 and 9 acute? Explain.

Choosing the Sine Law or the Cosine Law

1

Which law first? (a) ∠A, ∠B, a (b) a, b, c (c) b, c, ∠A (d) a, c, ∠C

2

Solve △ABC: ∠A = 40°, ∠C = 75°, b = 20 cm.

3

Why can't three angles alone determine a triangle's sides?

Real-World Problems: Surveying, Navigation and Construction

1

Two towers 20 km apart see a fire at 48° and 66° from the line joining them. How far is the fire from the first tower?

2

A ship sails 15 km on 050°, then 10 km on 170°. How far is it from its start?

3

A truss has a 6.0 m chord and rafters of 4.5 m and 4.0 m. Find the angle the 4.5 m rafter makes with the chord.

57°ABCc = 13 cmb = 9 cma = ?Known: two sides and theangle between them (SAS)→ cosine law first
The first question in the next section, drawn to scale: two sides and the angle between them.

Multi-Step Problems: Combining Triangles in 2-D and 3-D

1

A triangular park has sides 9 cm and 13 cm on a map with a 57° angle between them. Find the third side and (enrichment) the area on the map.

2

From points A and B, 100 m apart, the horizontal angles to the base F of a tower are 60° (at A) and 70° (at B). From A, the elevation to the top is 25°. Find the tower's height.

3

From a 50 m cliff, boats are seen at depression angles of 20° and 28°; the horizontal angle between them at the base is 55°. How far apart are the boats?

Check your understanding

Quiz

1.

In right △ABC with ∠C = 90°, which ratio is BC/AB?

2.

The angle of depression from a cliff top to a boat equals:

3.

What is the true bearing for N 70° W?

4.

Two observers on opposite sides of a tower: the distance between them is found by:

5.

The angle between a box's space diagonal and its base uses:

6.

Which technology formula verifies the sine law in a spreadsheet?

7.

You know two sides and the angle between them. Use:

8.

In acute △ABC, a = 16, b = 13, ∠A = 68°. ∠B is about:

9.

The largest angle of a triangle with sides 9, 10 and 12 is about:

10.

A boat sails on 050°, then turns to 170°. The interior angle at the turn is:

For parents and teachers

Use the review to find what to revisit:

  • Labelling opposite, adjacent and hypotenuse wrongly, or dividing when you should multiply → Lesson 1 (Finding Side Lengths).
  • Errors with sin⁻¹, cos⁻¹, tan⁻¹, or incomplete solutions → Lesson 2 (Finding Angles and Solving Right Triangles).
  • Angles measured from the wrong line, or bearings mixed up → Lesson 3 (Elevation, Depression and Bearings).
  • Adding when you should subtract in two-triangle problems → Lesson 4 (Two Right Triangles: Heights and Distances).
  • Altitude, navigation and river-width set-ups → Lesson 5 (Shared Sides, Navigation and Surveying).
  • Not seeing right angles in 3-D sketches → Lesson 6 (Right Triangles in Three Dimensions).
  • Pairing a side with the wrong angle → Lesson 7 (Discovering and Verifying the Sine Law).
  • Rearranging sin B/b = sin A/a incorrectly → Lesson 8 (Sine Law: Angles and Solving Triangles).
  • Order-of-operations errors in a² = b² + c² − 2bc cos A → Lesson 9 (Discovering and Verifying the Cosine Law).
  • Subtracting the wrong side's square, or missing brackets → Lesson 10 (Cosine Law: Angles and Solving Triangles).
  • Choosing the wrong law → Lesson 11 (Choosing the Sine Law or the Cosine Law).
  • Interior angles from bearings, or not answering the actual question → Lesson 12 (Real-World Problems).
  • Losing track in long or 3-D problems → Lesson 13 (Multi-Step Problems).

The next unit, Sine Functions, takes the sine ratio beyond 90° and turns it into a function that models repeating patterns such as tides and Ferris wheels.