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D. Trigonometric Functions · Trigonometric Ratios, Identities and Laws

Trigonometric Ratios, Identities and Laws: Unit Review

D1.1 D1.2 D1.3 D1.4 D1.5 D1.6 D1.7About 40 minutes
Learning goals
  • I can find exact and approximate trigonometric ratios for any angle from 0° to 360°.
  • I can solve for angles, use reciprocal ratios and prove simple identities.
  • I can solve oblique triangles, including the ambiguous case, and solve 2-D and 3-D problems.

Words to know

related acute angle
The acute angle between the terminal arm and the x-axis.
identity
An equation true for every value where both sides are defined.
ambiguous case
SSA information that may give zero, one or two triangles.

Let's review!

In this unit you built exact values from the special triangles, extended sine, cosine and tangent to every angle from 0° to 360° with x, y, r, the CAST rule and related acute angles, and found two angles with the same ratio. You met the reciprocal ratios csc, sec and cot, and proved identities. Then you took the sine law and cosine law into obtuse triangles, tamed the ambiguous case, and solved problems in two and three dimensions.

The next unit, Sinusoidal Functions, turns these ratios into functions and graphs them. 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, with your calculator in degree mode. Sketch every question: a quick drawing of the angle in standard position, or of the triangle, catches most sign and setup errors. Give exact answers when the angle is a special angle, and round other answers at the end only (lengths to one decimal place, angles to the nearest tenth of a degree unless asked otherwise). For each question you miss, write down why (wrong quadrant? wrong law? missed a second angle?) and go back to that lesson. Finish with the 10-question quiz.

Special trianglesexact values (lesson 1)Angles to 360°x, y, r and CAST (2 to 4)Reciprocal ratioscsc, sec, cot (5)Identitiesprove them (6, 7)Sine law, cosine lawoblique triangles (8, 9)Ambiguous caseSSA: 0, 1 or 2 (10)Problems in 2-D, 3-D(11, 12, 13)obtuse angles
How the unit fits together. The special triangles and CAST give values for any angle; those values power the identities and the laws for oblique triangles, which solve real problems in 2-D and 3-D.
Just for fun

You can now find the height of a tower across a river, the length of the longest rod in a room, and both possible triangles in the ambiguous case. That's a lot of power from three letters and their three reciprocals.

Exact Values: Special Triangles and Special Angles

1

Find the exact value of 2 sin 45° cos 45°.

2

Find the exact value of tan²30° + 1.

3

A 30-60-90 triangle has hypotenuse 12. Find the other two sides exactly.

Angles in Standard Position and the CAST Rule

1

The point (−7, 24) is on the terminal arm of θ. Find sin θ, cos θ and tan θ.

2

In which quadrant is cos θ > 0 and sin θ < 0?

3

tan θ = 3/4 and θ is in quadrant III. Find sin θ and cos θ.

Related Acute Angles and Exact Values from 0° to 360°

1

Find the exact value of cos 210°.

2

Find the exact value of tan 135°.

3

Find the exact value of sin 330° × cos 240°.

Solving for Angles: Two Angles with the Same Ratio

1

Solve cos θ = −1/2 for 0° ≤ θ ≤ 360°.

2

Solve sin θ = −0.3 for 0° ≤ θ ≤ 360°, to the nearest tenth of a degree.

3

Solve 2 tan θ − 2 = 0 for 0° ≤ θ ≤ 360°.

Reciprocal Trigonometric Ratios: Cosecant, Secant and Cotangent

1

Find the exact value of csc 60°.

2

Find the exact value of sec 135°.

3

Solve cot θ = 0.5 for 0° ≤ θ ≤ 360°, to the nearest tenth of a degree.

Trigonometric Identities: Pythagorean, Quotient and Reciprocal

1

Prove cos θ tan θ csc θ = 1.

2

sin θ = 0.28 and θ is in quadrant II. Find cos θ and tan θ.

3

Is tan θ + 1 = sec θ an identity?

Proving Trigonometric Identities: Strategies

1

Prove (1 + tan²θ)(1 − sin²θ) = 1.

2

Prove (sec θ − cos θ)/tan θ = sin θ.

3

Prove sin θ/(1 + cos θ) + (1 + cos θ)/sin θ = 2 csc θ.

The Sine Law in Oblique Triangles

1

In △ABC, ∠A = 125°, ∠B = 30° and c = 10 cm. Find a.

2

In △ABC, ∠C = 110°, c = 18 cm and b = 11 cm. Find ∠B.

3

Why does sin⁻¹ never give an obtuse angle, and what do you do about it?

The Cosine Law in Oblique Triangles

1

In △ABC, b = 6 m, c = 9 m and ∠A = 140°. Find a.

2

Find the largest angle of a triangle with sides 4, 7 and 9.

3

Without a calculator: is a triangle with sides 6, 8 and 11 acute or obtuse?

45°ACB₂B₁b = 12h ≈ 8.49a = 10h = 12 sin 45° ≈ 8.498.49 < 10 < 12two triangles
The first question in the next section: with ∠A = 45°, b = 12 and a = 10, side a can land at B₁ or B₂.

The Ambiguous Case of the Sine Law

1

∠A = 45°, b = 12 cm, a = 10 cm. Find all possible measures of ∠B.

2

∠A = 45°, b = 12 cm, a = 7 cm. How many triangles?

3

∠A = 100°, b = 5 cm, a = 8 cm. How many triangles?

Solving Problems in Two Dimensions

1

A boat travels 20 km on a bearing of 060°, then 15 km on a bearing of 150°. How far is it from the start, and on what bearing?

2

You know ∠B, a and b. Which law, and what do you check?

3

Pose a question about a triangular park with sides 80 m, 95 m and 130 m, and answer it.

Problems in Three Dimensions: Right Triangles

1

Find the space diagonal of a 2 × 3 × 6 box.

2

A 50 m tower is seen from P (due north of its foot) at an elevation of 40° and from Q (due east) at 30°. Find PQ.

3

Find the angle between the space diagonal of a 2 × 3 × 6 box and its 3 × 6 base.

Problems in Three Dimensions: Oblique Triangles

1

A and B are 100 m apart. The foot C of a tower makes ground angles ∠CAB = 55° and ∠CBA = 65°. From A, the angle of elevation of the top is 25°. Find the height.

2

A box is 4 cm by 4 cm by 2 cm. Find the angle at a bottom corner between the diagonals of the two 4 × 2 side faces.

3

Which triangle should an angle of elevation go in: the ground triangle or a vertical triangle?

Check your understanding

Quiz

1.

What is the exact value of sin 60°?

2.

What is tan 225°?

3.

Which angles between 0° and 360° have cos θ = −√3/2?

4.

What is csc 30°?

5.

Which equation is an identity?

6.

sin θ = 3/5 and θ is in quadrant II. What is cos θ?

7.

∠A = 30°, b = 10 and a = 6. How many triangles are possible?

8.

If ∠A is obtuse in △ABC, then:

9.

What is the space diagonal of a 2 × 3 × 6 box?

10.

You know two sides and the angle between them. Which tool do you use first?

For parents and teachers

Use the review to find what to revisit:

  • Special-angle values wrong, or radical arithmetic slips → Lesson 1 (Exact Values: Special Triangles and Special Angles).
  • Signs wrong, or confusion about x, y and r → Lesson 2 (Angles in Standard Position and the CAST Rule).
  • Right size but wrong sign, or β measured to the y-axis → Lesson 3 (Related Acute Angles).
  • Only one answer to an equation like sin θ = 0.6 → Lesson 4 (Two Angles with the Same Ratio).
  • Mixing up csc/sec, or pressing sin⁻¹ for a reciprocal → Lesson 5 (Reciprocal Ratios).
  • Proofs that move terms across the equals sign, or ± signs dropped → Lesson 6 (Trigonometric Identities).
  • Stuck partway through a longer proof → Lesson 7 (Proving Identities: Strategies).
  • Acute angle given where the angle is obtuse → Lesson 8 (The Sine Law in Oblique Triangles).
  • Order-of-operations errors in the cosine law → Lesson 9 (The Cosine Law in Oblique Triangles).
  • Missing the second triangle, or reporting an impossible one → Lesson 10 (The Ambiguous Case).
  • Bearings, choosing the right law, or multi-step setup → Lesson 11 (Problems in Two Dimensions).
  • Missing right angles in boxes, pyramids or towers → Lesson 12 (3-D: Right Triangles).
  • Mixing horizontal angles and angles of elevation → Lesson 13 (3-D: Oblique Triangles).

Looking ahead: the next unit, Sinusoidal Functions, graphs y = sin x and y = cos x and uses them to model periodic patterns such as tides and hours of daylight. These ideas continue in Grade 12 Advanced Functions (MHF4U).