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B. Trigonometric Functions · Radian Measure and Trigonometric Functions

Radian Measure and Trigonometric Functions: Unit Review

B1.1 B1.2 B1.3 B1.4 B2.1 B2.2 B2.3 B2.4 B2.5 B2.6 B2.7About 40 minutes
Learning goals
  • I can work fluently in radians: convert, find arc lengths, and find exact and approximate values of all six ratios.
  • I can sketch and describe the graphs of all six trigonometric functions with radian measures.
  • I can connect the equation y = a sin(k(x − d)) + c (or cosine) to its graph in both directions.
  • I can build sinusoidal models and use them to solve problems, including rates of change.

Words to know

radian
The central angle whose arc equals the radius; π rad = 180°.
period
2π/|k| for y = a sin(k(x − d)) + c; π for tan x and cot x.
vertical asymptote
A vertical line the graph approaches but never reaches, such as x = π/2 for tan x.
average rate of change
[f(b) − f(a)] ÷ (b − a), the slope of a secant line.

Let's review!

In this unit you met the radian, converted between radians and degrees, and found the ratios of radian angles with a calculator and exactly with the unit circle, including csc, sec and cot. You graphed sin x, cos x, tan x, csc x, sec x and cot x with radian measures, read amplitude, period and phase shift from equations, sketched transformed sinusoids, wrote equations from graphs and properties, and used models of Ferris wheels, tides, daylight and springs to answer questions about values, times and rates of change.

The next unit, Trigonometric Identities and Equations, uses all of this to prove identities and solve trigonometric equations in radians. This review has a few questions from every lesson. Try each one, then press Show answer.

How to use this review

Do one or two sections at a time with your calculator in radian mode (check: cos π = −1). Sketch a unit circle or a quick graph for every question; most errors are a wrong quadrant, a wrong sign or a phase shift read without factoring. Give exact answers for special angles. For each question you miss, write down why and go back to that lesson. Finish with the 10-question quiz.

Radianslessons 1, 2Ratios in radianslessons 3, 4, 5Graphs: sin, coslesson 6tan and reciprocalslessons 7, 8Transformationslessons 9, 10, 11Models and rateslessons 12, 13exact values
How the unit fits together. Radians feed the ratios; the ratios become the six graphs; transformations of the sinusoids lead to models and rates of change.
Just for fun

You now know two ways to measure every angle, six functions to apply to it, and four parameters to stretch the result. That's a lot of power from one circle with radius 1.

What Is a Radian? Measuring Angles with Arc Length

1

A circle has radius 8 cm. An arc of 20 cm subtends what central angle?

2

Find the exact arc length on a circle of radius 15 cm for a central angle of 2π/5.

3

Is an angle of 4 rad more or less than a half turn? Explain.

Converting Between Radians and Degrees

1

Convert 210° to radians exactly.

2

Convert 11π/12 to degrees.

3

Convert 3.5 rad to degrees, to the nearest tenth.

4

Find the angle between 0 and 2π that is coterminal with −7π/4.

Trigonometric Ratios of Angles in Radians, with Technology

1

Find sec 2.2 to four decimal places.

2

Find cot 1 to four decimal places.

3

The point (−8, 15) is on the terminal arm of θ, 0 ≤ θ < 2π. Find csc θ and θ to two decimal places.

Exact Values for Special Angles in Radians: The Unit Circle

1

Find the exact value of cos(2π/3).

2

Find the exact value of tan(11π/6).

3

Find the exact value of sin(5π/4) cos(π/6).

Exact Values of All Six Ratios, Including Coterminal and Quadrantal Angles

1

Find the exact value of csc(4π/3).

2

Find cot(3π/2).

3

Find the exact value of sec(−π/3).

4

Find tan(π/2), or explain why it does not exist.

The Graphs of y = sin x and y = cos x in Radians

1

List the zeros of y = sin x for −π ≤ x ≤ 2π.

2

On 0 ≤ x ≤ 2π, where is y = cos x increasing?

3

Write all the maximum points of y = cos x using n ∈ ℤ.

The Tangent Function

1

Write the equations of the asymptotes of y = tan x for −π ≤ x ≤ π.

2

What is the period of y = tan x, and why?

3

Given tan 0.3 ≈ 0.3093, find tan(π + 0.3) without a calculator.

Graphs of the Reciprocal Functions: csc x, sec x and cot x

1

State the range of y = csc x.

2

Write the equations of the asymptotes of y = sec x for 0 ≤ x ≤ 2π.

3

Find the zeros of y = cot x for 0 ≤ x ≤ π.

Amplitude, Period and Phase Shift in Radians

1

For y = −3 cos(4x + π) + 2, find the amplitude, period, phase shift and range.

2

Find k > 0 for a period of 5π.

3

What is the period of y = 7 sin(πx/10)?

Sketching Transformed Sinusoidal Functions

1

Write the mapping rule for y = 2 sin(3(x − π/6)) − 1.

2

List the five key points of one cycle of that function.

3

State its maximum points, using n ∈ ℤ.

01234567891011121312345678xy(1, 7)(4, 3)(7, 7)(10, 3)
A sinusoidal graph for the next section. Read the maximum, minimum and period from the marked points.

Writing the Equation of a Sinusoidal Function

1

Write a cosine equation for the graph above.

2

A graph has a maximum at (π/4, 6) and the next minimum at (3π/4, 2). Write a cosine equation.

3

Write a sine equation for the same graph, with no phase shift.

Modelling Periodic Phenomena with Sinusoidal Functions

1

A Ferris wheel has radius 10 m, its centre is 12 m above the ground, and it turns once every 40 s. A rider boards at the bottom at t = 0. Write a model.

2

Find the rider's height at t = 5 s exactly.

3

In D(t) = 3.2 sin((2π/365)(t − 80)) + 12.2, what do 3.2 and 80 mean?

Solving Problems with Sinusoidal Models: Times, Intervals and Rates of Change

1

For h(t) = −10 cos(πt/20) + 12, when is the rider first 17 m above the ground?

2

Find the average rate of change of h from t = 0 to t = 20 s.

3

At what times in the first rotation is the height changing fastest? Explain.

Check your understanding

Quiz

1.

What is 3π/4 in degrees?

2.

A circle has radius 6 cm. What is the arc length for a central angle of π/3?

3.

What is the exact value of sin(4π/3)?

4.

What is the exact value of sec(π/4)?

5.

What is the period of y = tan x?

6.

Where does y = sec x have vertical asymptotes?

7.

What is the period of y = 5 sin(πx/4) − 1?

8.

What is the phase shift of y = sin(3x − π/2)?

9.

Which equation has a maximum at (0, 9), a minimum at (π, 1) and period 2π?

10.

What is the average rate of change of y = 3 sin x + 2 over 0 ≤ x ≤ π/2?

For parents and teachers

Use the review to find what to revisit:

  • Using a = rθ with degrees, or unsure what a radian is → Lesson 1 (What Is a Radian?).
  • Conversion slips or unsimplified fractions of π → Lesson 2 (Converting Between Radians and Degrees).
  • Wrong calculator mode, or sin⁻¹ used for csc → Lesson 3 (Ratios of Angles in Radians, with Technology).
  • Exact values with the wrong sign or the wrong special angle → Lesson 4 (The Unit Circle).
  • Reciprocal ratios, "undefined" versus 0, or angles outside 0 to 2π → Lesson 5 (All Six Ratios).
  • Zeros, maxima or increasing intervals of sin and cos → Lesson 6 (Graphs of sin x and cos x).
  • Asymptotes or period of tan x → Lesson 7 (The Tangent Function).
  • csc, sec or cot graphs → Lesson 8 (Reciprocal Functions).
  • Phase shift read without factoring, or period written as 2πk → Lesson 9 (Amplitude, Period and Phase Shift).
  • Key points wrongly placed or spaced → Lesson 10 (Sketching Transformed Sinusoids).
  • Equations that do not match the graph → Lesson 11 (Writing Equations).
  • Models with the wrong k, sign or units → Lesson 12 (Modelling Periodic Phenomena).
  • Only one solution per cycle, or rate-of-change errors → Lesson 13 (Solving Problems and Rates of Change).

Looking ahead: the next unit, Trigonometric Identities and Equations, proves identities such as sin(π/2 − x) = cos x and solves equations like 2 sin²x − sin x − 1 = 0 on 0 ≤ x ≤ 2π. The unit Rates of Change takes the idea of average and instantaneous rates much further.