MATH 130 · Module 11

Trigonometric Functions of Any Angle

Concepts, worked examples, student practice, and synthesis

55 minutes · Offline capable

Today’s Learning Roadmap

1
find reference angles in degrees and radians
2
determine trigonometric signs and quadrants
3
find exact trigonometric values from points and reference angles
4
apply the trigonometric triangle-area formula

Activate · 0.5 minSource: 1

Predict: find reference angles in degrees and radians

Predict first

Why is a reference angle always acute?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

find reference angles in degrees and radians

Formula focus

Reference angles are measured to the nearest x-axis.
First find a coterminal angle when the original lies outside one rotation.

Explain · 2.5 minSource: 2, 3

Model the Skill: find reference angles in degrees and radians

$150^\circ$ has reference angle $30^\circ$; $310^\circ$ has $50^\circ$.

$-120^\circ$ is coterminal with $240^\circ$, so its reference angle is $60^\circ$.

$7\pi/4$ has reference angle $\pi/4$.

Model · 3.0 minSource: 5

You Try: find reference angles in degrees and radians

Student practice

Find the reference angle for 580°.

Feedback and solution

$580-360=220^\circ$, so the QIII reference angle is 40°.

Practice · 3.0 minSource: 6, 7

Check and Diagnose: find reference angles in degrees and radians

Error analysis

Do not measure to the y-axis.
Use π, not 180°, when working in radians.

Self-check

What evidence confirms the setup, sign, unit, or magnitude?

Feedback · 2.0 minSource: 8

Pacing checkpoint

Make It Stick: find reference angles in degrees and radians

State the quadrant formulas for reference angles.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: 9

Predict: determine trigonometric signs and quadrants

Predict first

If tangent is positive and cosine is negative, which quadrant is possible?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

determine trigonometric signs and quadrants

ASTC summarizes positive functions by quadrant.

Signs follow from x, y, and positive r.

Explain · 2.5 minSource: 10

Model the Skill: determine trigonometric signs and quadrants

Determine the quadrant if $\tan\theta>0$ and $\cos\theta<0$.

Tangent is positive in Quadrants I and III.
Cosine is negative in Quadrants II and III.
The intersection is Quadrant III.

Model · 3.0 minSource: Authored

You Try: determine trigonometric signs and quadrants

Student practice

Determine the quadrant if $\sin\theta>0$ and $\sec\theta<0$.

Feedback and solution

Quadrant II.

Practice · 3.0 minSource: 11

Check and Diagnose: determine trigonometric signs and quadrants

Use the intersection of sign conditions.

Reciprocal functions share the sign of their partner.

Feedback · 2.0 minSource: Authored

Pacing checkpoint

Make It Stick: determine trigonometric signs and quadrants

Explain ASTC using the signs of x and y rather than memorization.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: Authored

Predict: find exact trigonometric values from points and reference angles

Predict first

What does r represent for a point $(x,y)$ on a terminal side?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

find exact trigonometric values from points and reference angles

Concept

$r=\sqrt{x^2+y^2}$.
Use the reference angle for magnitude and the quadrant for sign.

Explain · 2.5 minSource: 4, 12

Model the Skill: find exact trigonometric values from points and reference angles

For $P=(-3,4)$, calculate r and all six trigonometric values.

$r=\sqrt{(-3)^2+4^2}=5$.
$\sin\theta=4/5$, $\cos\theta=-3/5$, $\tan\theta=-4/3$.
$\csc\theta=5/4$, $\sec\theta=-5/3$, $\cot\theta=-3/4$.

Model · 3.0 minSource: 13

You Try: find exact trigonometric values from points and reference angles

Student practice

Find exact trig values for $P=(-2,-5)$ and evaluate $\sin240^\circ$.

Feedback and solution

For $P=(-2,-5)$, $r=\sqrt{29}$; $\sin=-5/\sqrt{29}$, $\cos=-2/\sqrt{29}$, $\tan=5/2$.
$\csc=-\sqrt{29}/5$, $\sec=-\sqrt{29}/2$, $\cot=2/5$; $\sin240^\circ=-\sqrt3/2$.

Practice · 3.0 minSource: 14, 15

Check and Diagnose: find exact trigonometric values from points and reference angles

Error analysis

r is always positive.
Rationalize denominators only when required by course convention.

Self-check

What evidence confirms the setup, sign, unit, or magnitude?

Feedback · 2.0 minSource: 16

Pacing checkpoint

Make It Stick: find exact trigonometric values from points and reference angles

Describe the point-to-ratios workflow in three steps.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: 17

Predict: apply the trigonometric triangle-area formula

Predict first

Why must the known angle be included between the two known sides?

Commit to a response before discussion.

Activate · 1.0 minSource: Authored

apply the trigonometric triangle-area formula

Concept

$A=\tfrac12bc\sin A$ uses two sides and their included angle.
The formula applies to nonright triangles.

Explain · 2.5 minSource: 18

Model the Skill: apply the trigonometric triangle-area formula

Find the area when $A=76^\circ$, $b=34$, and $c=21$.

$A_{triangle}=\tfrac12bc\sin A$.
$A_{triangle}=\tfrac12(34)(21)\sin76^\circ\approx346.40$.
The result is in square units.

Model · 3.0 minSource: 19

You Try: apply the trigonometric triangle-area formula

Student practice

Find the area when sides 12 and 15 include a 40° angle.

Feedback and solution

$A=\tfrac12(12)(15)\sin40^\circ\approx57.85$.

Practice · 3.0 minSource: Authored

Check and Diagnose: apply the trigonometric triangle-area formula

Error analysis

Match each side pair with its included angle.
Square units are required.

Self-check

What evidence confirms the setup, sign, unit, or magnitude?

Feedback · 2.0 minSource: 20

Pacing checkpoint

Make It Stick: apply the trigonometric triangle-area formula

Explain how the diagram identifies the correct sine angle.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Synthesize · 0.5 minSource: Authored

Session Summary and Exit Ticket

State the most important idea from today without looking at your notes.
Complete one representative setup and identify the step most likely to cause an error.
Write one question that should be answered before the next assessment.

Exit response

Answer in complete mathematical sentences and identify one remaining question.

Feedback and solution

Use the objective roadmap to identify any skill that still needs deliberate practice.

Synthesize · 1.5 minSource: 21