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B. Trigonometric Functions · Trigonometric Identities and Equations

Trigonometric Identities and Equations: Unit Review

B3.1 B3.2 B3.3 B3.4About 40 minutes
Learning goals
  • I can recognize and use equivalent trigonometric expressions, compound angle formulas and double angle formulas.
  • I can prove identities and disprove false ones with a counterexample.
  • I can solve linear and quadratic trigonometric equations for 0 ≤ x ≤ 2π and use them to solve problems.

Words to know

compound angle formula
An identity for a ratio of a ± b, such as sin(a + b) = sin a cos b + cos a sin b.
identity
An equation true for every value in the domain; a single counterexample shows an equation is not one.
related acute angle
The angle α between the terminal arm and the x-axis, used with CAST to find every solution.

Let's review!

In this unit you learned to see when two trigonometric expressions are equivalent (cofunctions, symmetry and shifts), followed the development of the compound angle formulas and used them, with the double angle formulas, to find exact values such as cos(π/12) = (√6 + √2)/4. You learned that an identity is true for every value in its domain, that one counterexample disproves a false one, and how to prove identities step by step. Finally, you solved linear and quadratic trigonometric equations in radians on 0 ≤ x ≤ 2π, including equations that need an identity first, and used them to answer real questions about Ferris wheels, tides and daylight.

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 radian mode. Keep your identity sheet beside you at first, then try a section without it. For equations, sketch the four-quadrant diagram (α, π − α, π + α, 2π − α) and list every solution in 0 ≤ x ≤ 2π, including endpoints. For proofs, write L.S. and R.S. and never move terms across the equals sign. For each question you miss, write down why (wrong sign? wrong form of cos 2x? lost a solution by dividing?) and go back to that lesson. Finish with the 10-question quiz.

Equivalentexpressionslessons 1–2Compound angleformulaslessons 3–5Double angleformulaslesson 6Provingidentitieslessons 7–9Linearequationslessons 10–11Quadraticequationslessons 12–13Solvingproblemslesson 14identities make equations solvable
How the unit fits together. Equivalent expressions lead to the compound and double angle formulas; those formulas power the proofs, and the same identities turn hard equations into solvable ones.
Just for fun

You can now find the exact value of cos(π/12), prove that sin 3x = 3 sin x − 4 sin³x, and tell a fishing boat when the tide will let it home. Not bad for a few pages of sines and cosines.

Equivalent Expressions: Cofunction Identities

1

Write cos(2π/9) as the sine of another angle.

2

Given sin(π/7) ≈ 0.4339, find cos(5π/14).

3

Simplify sin(π/2 − x) tan x.

Equivalent Expressions from Transformations and Symmetry

1

Simplify cos(π + x) + cos(−x).

2

Find the exact value of sin(5π/3).

3

Describe the transformation that shows cos(x − π/2) = sin x, and check it at x = π/6.

Compound Angle Formulas for Cosine

1

Find the exact value of cos(5π/12).

2

Simplify cos 3x cos x + sin 3x sin x.

3

cos a = 4/5 with a in quadrant I, and sin b = 12/13 with b in quadrant II. Find cos(a − b).

Compound Angle Formulas for Sine

1

Find the exact value of sin(7π/12).

2

Write sin x cos(π/3) − cos x sin(π/3) as a single sine.

3

Use a compound angle formula to simplify sin(x + 3π/2).

Compound Angle Formulas for Tangent

1

Find the exact value of tan(π/12).

2

a and b are acute with tan a = 1/2 and tan b = 1/3. Find a + b.

3

Simplify (tan 5x − tan 2x)/(1 + tan 5x tan 2x).

Double Angle Formulas

1

cos x = −3/5 with π/2 < x < π. Find sin 2x and cos 2x.

2

Find the exact value of cos²(π/8) − sin²(π/8).

3

True or false: cos 2x = 2 cos x for all x.

Identities, Equations and Counterexamples

1

Find a counterexample to tan 2x = 2 tan x.

2

State the restrictions for cot x = cos x/sin x.

3

Is cos(x − π) = −cos x an identity? Justify.

Proving Identities: Reciprocal, Quotient and Pythagorean

1

Prove sin x cot x = cos x.

2

Prove (1 − sin²x)(1 + tan²x) = 1.

3

Prove sin²x/(1 − cos x) = 1 + cos x.

Proving Identities with Compound and Double Angle Formulas

1

Prove (sin x − cos x)² = 1 − sin 2x.

2

Prove (1 + cos 2x)/sin 2x = cot x.

3

Prove sin(x + π/6) + cos(x + π/3) = cos x.

Solving Linear Trigonometric Equations in Radians

1

Solve 2 cos x + 1 = 0 for 0 ≤ x ≤ 2π.

2

Solve tan x = −√3 for 0 ≤ x ≤ 2π.

3

Solve 5 sin x + 3 = 0 for 0 ≤ x ≤ 2π, to two decimal places.

Linear Equations with Multiple and Shifted Angles

1

Solve sin 2x = √2/2 for 0 ≤ x ≤ 2π.

2

Solve cos(x − π/6) = 0 for 0 ≤ x ≤ 2π.

3

Solve tan(x/2) = 1 for 0 ≤ x ≤ 2π.

Solving Quadratic Trigonometric Equations by Factoring

1

Solve 2sin²x − 1 = 0 for 0 ≤ x ≤ 2π.

2

Solve 2cos²x − 3 cos x + 1 = 0 for 0 ≤ x ≤ 2π.

3

Solve tan²x − 2 tan x = 0 for 0 ≤ x ≤ 2π (two decimal places where needed).

xπ/2π3π/22π1−1π/65π/6solid: y = sin 2x; dashed: y = cos x
A graph check for lesson 13, Example 2: y = sin 2x and y = cos x meet at π/6, π/2, 5π/6 and 3π/2, so sin 2x = cos x has four solutions in 0 ≤ x ≤ 2π.

Solving Trigonometric Equations Using Identities

1

Solve sin 2x + sin x = 0 for 0 ≤ x ≤ 2π.

2

Solve 2sin²x + cos x − 1 = 0 for 0 ≤ x ≤ 2π.

3

Solve cos 2x = sin x for 0 ≤ x ≤ 2π.

Solving Problems with Trigonometric Equations

1

A Ferris wheel rider's height is h(t) = 12 − 10 cos(πt/3) metres at t minutes, 0 ≤ t ≤ 6. When is the rider at 12 m?

2

A spring's displacement is d(t) = 4 cos(2πt) cm, 0 ≤ t ≤ 1. When is d = 4 cm?

3

In the model D(t) = 12.2 + 3.5 sin(2π(t − 80)/365), on which day is daylight greatest?

Check your understanding

Quiz

1.

Which expression is equivalent to sin x for every x?

2.

What is the exact value of cos(π/12)?

3.

Which formula is correct for every x?

4.

Which value of x is a counterexample to tan 2x = 2 tan x?

5.

Which formula is correct?

6.

Which step is allowed when proving an identity?

7.

What are the solutions of 2 cos x − 1 = 0 for 0 ≤ x ≤ 2π?

8.

How many solutions does sin 2x = 1/2 have for 0 ≤ x ≤ 2π?

9.

What are the solutions of 2sin²x − sin x − 1 = 0 for 0 ≤ x ≤ 2π?

10.

To solve cos 2x = sin x, which form of cos 2x should you use?

For parents and teachers

Use the review to find what to revisit:

  • Mixing up complements, or pairing the wrong cofunctions → Lesson 1 (Cofunction Identities).
  • Sign errors with π − x, π + x, −x, or shifting a graph the wrong way → Lesson 2 (Transformations and Symmetry).
  • Wrong sign in the middle of cos(a ± b), or choosing special angles that do not add up → Lesson 3 (Compound Angle Formulas for Cosine).
  • Mixing the sine and cosine patterns, or losing a sign from the quadrant → Lesson 4 (Compound Angle Formulas for Sine).
  • Sign errors in the tangent denominator, or not rationalizing → Lesson 5 (Compound Angle Formulas for Tangent).
  • Thinking sin 2x = 2 sin x, or using the wrong form of cos 2x → Lesson 6 (Double Angle Formulas).
  • Calling an equation an identity after one test, or ignoring restrictions → Lesson 7 (Identities, Equations and Counterexamples).
  • Moving terms across the equals sign, or fraction errors in proofs → Lesson 8 (Proving Identities: Reciprocal, Quotient and Pythagorean).
  • Stuck on proofs with 2x or x + y → Lesson 9 (Proving Identities with Compound and Double Angle Formulas).
  • Only one solution found, or calculator in degree mode → Lesson 10 (Solving Linear Equations in Radians).
  • Missing solutions for sin 2x or tan 3x, or wrong interval for θ → Lesson 11 (Multiple and Shifted Angles).
  • Dividing by sin x, or keeping sin x = −2 → Lesson 12 (Quadratic Equations by Factoring).
  • Two ratios left in the equation, or extraneous answers after squaring → Lesson 13 (Equations Using Identities).
  • Answers left in radians instead of minutes or days, or not interpreted → Lesson 14 (Solving Problems).

Looking ahead: the next unit, Polynomial Functions and Equations, turns to strand C. Trigonometric identities return in Combining Functions and Modelling (sums and products of functions) and in Calculus and Vectors (MCV4U), where derivatives of sin x and cos x rely on the compound angle formulas.