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Quiz Review

Trigonometric Functions

Sections 4.1 -- 4.5

Goals

Learning Targets

  • ✓ Identify which trig function to use based on given information
  • ✓ Use reference angles and quadrant analysis to evaluate trig functions
  • ✓ Apply the terminal side method to find all six trig values
  • ✓ Recognize when a problem requires the unit circle vs. calculator
Reference

The Quiz Strategy Guide

"Find \(\sin 225^\circ\)" (special angle) → Draw reference angle → Use unit circle
"Find \(\tan\theta\), given \(\sin\theta\) & QII" → Sketch in quadrant → Pythagorean identity
"Terminal side through \((-3,\,4)\)" → Find \(r\) → Use \(x/r,\;y/r\) definitions
"Evaluate \(\csc 137^\circ\)" → Check calculator mode → Reciprocal after
1
Unit Circle Mastery

WITHOUT a calculator, find exact values

Cycle 1

Problem Set A: Find the Exact Value

a) \(\sin 30^\circ\)
b) \(\cos 240^\circ\)
c) \(\tan\dfrac{5\pi}{4}\)
d) \(\sin\dfrac{11\pi}{6}\)
e) \(\cot 3\pi\)
f) \(\sec\!\left(-\dfrac{\pi}{3}\right)\)
Write your reference angle for each answer.
Solutions

Unit Circle & Reference Angles

I II III IV
a) \(\sin 30^\circ\)
QI — Ref: \(30^\circ\)
\(= \dfrac{1}{2}\)
b) \(\cos 240^\circ\)
QIII (cos −) — Ref: \(60^\circ\)
\(= -\dfrac{1}{2}\)
c) \(\tan\frac{5\pi}{4}\;(225^\circ)\)
QIII (tan +) — Ref: \(45^\circ\)
\(= 1\)
d) \(\sin\frac{11\pi}{6}\;(330^\circ)\)
QIV (sin −) — Ref: \(30^\circ\)
\(= -\dfrac{1}{2}\)
e) \(\cot 3\pi\;(540^\circ)\)
Negative x-axis — Ref: \(0^\circ\)
\(= \text{undefined}\)
f) \(\sec\!\left(-\frac{\pi}{3}\right)\;(-60^\circ)\)
QIV (cos +) — Ref: \(60^\circ\)
\(= 2\)
Click any problem to see the angle on the unit circle!
Check

Quick Check — Trouble Spots

Which problem did you skip or guess on?

Click to mark your trouble spots:

These are the problems to review before the quiz!

2
Quadrant Analysis

Eliminate impossibilities → Find the overlap

Cycle 2

Problem Set B: Find the Quadrant

I II III IV sin+, cos+ tan+ sin+, cos- tan- sin-, cos- tan+ sin-, cos+ tan-
Problem 1
\(\sin\theta > 0\) and \(\tan\theta < 0\)
→ Find the quadrant
Problem 2
\(\cos\theta < 0\) and \(\sin\theta > 0\)
→ Find the quadrant
Problem 3
\(\sec\theta = 4\) and \(0 \leq \theta \leq \frac{\pi}{2}\)
→ Find the quadrant
Problem 4
\(\sin\theta = \frac{3}{5}\) and \(\cos\theta = -\frac{4}{5}\)
→ Find sin, cos, tan, and the quadrant
Use the quadrant diagram to eliminate impossibilities, then find the overlap!
Cycle 2

Check Your Answers

Problem 1
\[\textbf{QII}\]
sin+ and tan- only overlap in QII
Problem 2
\[\textbf{QII}\]
cos- and sin+ only overlap in QII
Problem 3
\[\textbf{QI}\]
Already restricted by \(0 \leq \theta \leq \frac{\pi}{2}\)
Problem 4
\[\textbf{QII}, \quad \tan\theta = -\frac{3}{4}\]
sin+ and cos- → QII, \(\tan = \frac{3/5}{-4/5} = -\frac{3}{4}\)
3
Terminal Side Setup

When terminal side passes through a point \((x, y)\)

Worked Example

Terminal Side through \((-3, 4)\)

Find all six trig values when the terminal side of \(\theta\) passes through \((-3, 4)\).
x = -3 y = 4 r = 5 (-3, 4)
Given Point
\[x = -3, \quad y = 4 \quad \Rightarrow \quad \text{Quadrant II}\]
Step 1 — Find r
\[r = \sqrt{x^2 + y^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]
Step 2 — Apply definitions
\[\sin\theta = \frac{y}{r}, \quad \cos\theta = \frac{x}{r}, \quad \tan\theta = \frac{y}{x}\]
Step 3 — All six values
\[\sin\theta = \frac{4}{5} \qquad \csc\theta = \frac{5}{4}\] \[\cos\theta = -\frac{3}{5} \qquad \sec\theta = -\frac{5}{3}\] \[\tan\theta = -\frac{4}{3} \qquad \cot\theta = -\frac{3}{4}\]
Your Turn

Problem Set C

Terminal side passes through the point \((5, -12)\) — Quadrant IV

On your paper, find:

1) \(r\)  (use \(r = \sqrt{x^2 + y^2}\))
2) \(\sin\theta\)  (use \(\sin = y/r\))
3) \(\csc\theta\)  (reciprocal of sin)
4) \(\cot\theta\)  (use \(\cot = x/y\))
Hint: 5, 12, 13 is a Pythagorean triple!
Solutions

Check Your Setup — Point \((5, -12)\)

5 10 5 -5 -10 x = 5 y = -12 r = 13 (5, -12)
Given Point
\[x = 5, \quad y = -12 \quad \Rightarrow \quad \text{Quadrant IV}\]
Step 1 — Find r
\[r = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13\]
Step 2 — Apply definitions
\[\sin\theta = \frac{y}{r}, \quad \csc\theta = \frac{r}{y}, \quad \cot\theta = \frac{x}{y}\]
Step 3 — All six values
\[\sin\theta = -\frac{12}{13} \qquad \csc\theta = -\frac{13}{12}\] \[\cos\theta = \frac{5}{13} \qquad \sec\theta = \frac{13}{5}\] \[\tan\theta = -\frac{12}{5} \qquad \cot\theta = -\frac{5}{12}\]
4
Arc Length

Using \(s = r\theta\) to find arc length

Cycle 4

Problem Set D: Arc Length

Use the formula \(s = r\theta\)  (\(\theta\) must be in radians!)
a) \(r = 8\) cm, \(\theta = \frac{\pi}{3}\) rad → find \(s\)
b) \(r = 12\) in, \(\theta = 150^\circ\) → find \(s\)
c) \(s = 20\) m, \(r = 5\) m → find \(\theta\) (radians)
d) \(s = 15\pi\) ft, \(\theta = \frac{2\pi}{3}\) rad → find \(r\)
e) Wheel: 3 full rotations, \(r = 2\) ft → distance?
Remember: Convert degrees to radians first! \(\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}\)
Worked Example

Arc Length: \(r = 6\), \(\theta = \frac{\pi}{3}\)

Find the arc length when \(r = 6\) cm and \(\theta = \frac{\pi}{3}\) radians.
r = 6 \(\frac{\pi}{3}\) s = 2\(\pi\)
Given Information
\[r = 6 \text{ cm}, \quad \theta = \frac{\pi}{3} \text{ rad}\] \[\text{Find: arc length } s\]
Step 1 — Apply the formula
\[s = r\theta = 6 \cdot \frac{\pi}{3}\]
Step 2 — Simplify
\[s = \frac{6\pi}{3} = 2\pi \text{ cm} \approx 6.28 \text{ cm}\]
Key Insight
\[\theta = \frac{\pi}{3} \text{ is } \frac{1}{6} \text{ of a full circle } (2\pi)\] \[\text{Circumference} = 2\pi r = 12\pi, \quad s = \frac{1}{6} \cdot 12\pi = 2\pi \;\checkmark\]
Cycle 4

Check Your Answers

Problem a
\[s = \frac{8\pi}{3} \approx 8.38 \text{ cm}\]
Direct: \(s = r\theta = 8 \cdot \frac{\pi}{3}\)
Problem b
\[s = 10\pi \approx 31.42 \text{ in}\]
\(150^\circ = \frac{5\pi}{6}\) rad, then \(s = 12 \cdot \frac{5\pi}{6}\)
Problem c
\[\theta = 4 \text{ radians}\]
\(\theta = \frac{s}{r} = \frac{20}{5} = 4\)
Problem d
\[r = 22.5 \text{ ft}\]
\(r = \frac{s}{\theta} = \frac{15\pi}{2\pi/3} = \frac{45}{2}\)
Problem e
\[s = 12\pi \approx 37.70 \text{ ft}\]
3 rotations = \(6\pi\) rad, \(s = 2 \cdot 6\pi\)
5
Right Triangle Applications

Multi-step word problems with real-world scenarios

Cycle 5

Problem Set E: Word Problems

Draw a diagram for each scenario. Identify all right triangles.

Problem 1: Two Observation Points

Two people stand 50 meters apart on level ground, both looking at the top of a building. Person A (closer) measures an angle of elevation of \(60^\circ\). Person B measures an angle of elevation of \(30^\circ\). Find the height of the building.

Problem 2: Ship Navigation

A ship leaves port and sails 40 km on a bearing of N \(60^\circ\) E. It then changes course and sails 30 km on a bearing of N \(30^\circ\) W. How far is the ship from port?
Hint: Break each leg into north and east components.

Problem 3: Sliding Ladder

A 20-foot ladder leans against a wall at an angle of \(75^\circ\) with the ground. The base slides out 4 feet farther from the wall. What is the new angle with the ground? How far did the top slide down?

Set up equations before solving. Label all known and unknown values on your diagram!
Worked Example

Two Observation Points

Two people 50 m apart. Person A: \(60^\circ\) elevation. Person B: \(30^\circ\) elevation. Find building height \(h\).
h = ? Person A Person B 50 m 60° 30° x
Given Information
\[\text{50 m apart, } \alpha_A = 60^\circ, \; \alpha_B = 30^\circ\] \[\text{Find: height } h\]
Step 1 — Equation from Person A
\[\text{Let } x = \text{distance from building to A}\] \[\tan(60^\circ) = \frac{h}{x} \quad \Rightarrow \quad h = x\sqrt{3}\]
Step 2 — Equation from Person B
\[\tan(30^\circ) = \frac{h}{x + 50} \quad \Rightarrow \quad h = \frac{x + 50}{\sqrt{3}}\]
Step 3 — Solve the system
\[x\sqrt{3} = \frac{x + 50}{\sqrt{3}}\] \[3x = x + 50 \quad \Rightarrow \quad 2x = 50 \quad \Rightarrow \quad x = 25 \text{ m}\] \[h = 25\sqrt{3} \approx 43.3 \text{ meters}\]
Cycle 5

Check Your Answers

Problem 1
\[h = 25\sqrt{3} \approx 43.3 \text{ m}\]
Set \(x\sqrt{3} = \frac{x+50}{\sqrt{3}}\), solve \(x = 25\)
Problem 2
\[\approx 61.0 \text{ km from port}\]
E: \(40\sin 60^\circ - 30\sin 30^\circ \approx 19.6\)
N: \(40\cos 60^\circ + 30\cos 30^\circ \approx 46.0\)
\(d = \sqrt{E^2 + N^2}\)
Problem 3
\[\theta \approx 60.1^\circ, \quad \text{slid down} \approx 3.1 \text{ ft}\]
New base \(= 20\cos 75^\circ + 4 \approx 9.18\) ft
\(\cos^{-1}\!\left(\frac{9.18}{20}\right) \approx 60.1^\circ\)
Wrap Up

Practice Time!

1
Review All Five Cycles
Unit circle, quadrant analysis, terminal side, arc length, and right triangle word problems
2
Practice Delta Math Application Problems
Work through the assigned problems including arc length and multi-step word problems
3
Solving Angles on Google Slides
Solve the Free Response angle-solving problems on Google Slides. Take a picture and upload your answers to the slides and turn in.
Remember: Use the Quiz Strategy Guide to choose your approach for each problem!
Post in Chat

Async Work Checklist

  • Complete terminal side problem (Problem Set C) 8 min
  • Arc length practice problems (Problem Set D) 10 min
  • Word problem diagrams (Problem Set E) Offline 15 min
  • Unit circle video (ONLY if you marked trouble spots) 8 min
  • Draw reference angle diagrams Offline 10 min
  • Quadrant analysis with full work 12 min
  • Conversions practice (degrees / radians, DMS) 10 min
  • Applications walkthrough 7 min
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