Unit 6 Worksheet 13: Finding Reference Angles - A math worksheet with 20 problems requiring students to calculate the reference angle for various given angles in degrees and radians.

Explanation:
We are asked to find the reference angle θ′ for each given angle θ.

A reference angle is the smallest positive acute angle (between 0° and 90°, or 0 and π/2 radians) that the terminal side of the angle makes with the x-axis. It’s always positive and ≤ 90° (or π/2 rad).

Steps to find reference angle:
1. First, find the coterminal angle between 0° and 360° (for degrees) or between 0 and 2π (for radians), by adding or subtracting multiples of 360° or 2π.
2. Then determine which quadrant the angle lies in:
- QI (0° to 90°): reference angle = angle itself
- QII (90° to 180°): reference angle = 180° − angle
- QIII (180° to 270°): reference angle = angle − 180°
- QIV (270° to 360°): reference angle = 360° − angle
For radians, replace 180° with π, 360° with 2π.

Let’s go through each part carefully.

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A) θ = 57°
Already between 0° and 90° → QI
→ Reference angle = 57°

B) θ = 113°
Between 90° and 180° → QII
→ Reference angle = 180° − 113° = 67°

C) θ = 7π/6
Convert to degrees if helpful: π = 180°, so 7π/6 = 7×30 = 210°
210° is in QIII (180°–270°)
→ Reference angle = 210° − 180° = 30°
In radians: 7π/6 − π = π/6
→ Reference angle = π/6

D) θ = 5π/3
5π/3 = 5×60 = 300° → QIV
Reference angle = 360° − 300° = 60°
In radians: 2π − 5π/3 = (6π/3 − 5π/3) = π/3
→ Reference angle = π/3

E) θ = −2π/3
Negative angle → add 2π to get coterminal:
−2π/3 + 2π = −2π/3 + 6π/3 = 4π/3
4π/3 = 240° → QIII
Reference angle = 4π/3 − π = π/3
→ π/3

F) θ = −230°
Add 360°: −230° + 360° = 130° → QII
Reference angle = 180° − 130° = 50°

G) θ = 300°
QIV
Reference angle = 360° − 300° = 60°

H) θ = 2.3
This is in radians (no degree symbol).
Compare to π ≈ 3.1416, π/2 ≈ 1.5708
2.3 is between π/2 and π → QII
Reference angle = π − 2.3 ≈ 3.1416 − 2.3 = 0.8416 rad
But since the question likely expects exact or simplified form, and 2.3 is decimal, we keep it as π − 2.3 (or approximate). However, in most textbook problems like this, if it's a decimal like 2.3, they expect the answer as π − 2.3 (since no nice fraction). Let’s hold and check others — maybe all answers are expected in simplest exact form or degrees/radians as appropriate.

Wait — let’s double-check: The problem says “find the reference angle θ′”, but doesn’t specify format. Since inputs mix degrees and radians, outputs should match type (degrees for degree inputs, radians for radian inputs), and be simplified.

So for H: θ = 2.3 rad → reference angle = π − 2.3 rad
We can leave as π − 2.3 (exact) or approximate 0.84 rad. But since others are exact, better to write π − 2.3.

But maybe the worksheet expects numeric? Let’s see other decimal ones:

I) θ = 280°
QIV → 360° − 280° = 80°

J) θ = 1.2 rad
1.2 < π/2 (≈1.57) → QI
Reference angle = 1.2 rad

K) θ = 420°
Subtract 360°: 420 − 360 = 60° → QI
Reference angle = 60°

L) θ = −60°
Add 360°: 300° → QIV
Reference angle = 360° − 300° = 60°

M) θ = −2 rad
Add 2π: −2 + 2π ≈ −2 + 6.283 = 4.283 rad
4.283 > π (3.1416) and < 3π/2 (4.712) → QIII
Reference angle = (−2 + 2π) − π = π − 2
Because: coterminal = 2π − 2, then subtract π → (2π − 2) − π = π − 2
So reference angle = π − 2 rad
(Positive since π ≈ 3.14 > 2)

N) θ = 100°
QII → 180° − 100° = 80°

O) θ = −4π/3
Add 2π: −4π/3 + 6π/3 = 2π/3
2π/3 = 120° → QII
Reference angle = π − 2π/3 = π/3
→ π/3

P) θ = 11π/6
11π/6 = 330° → QIV
Reference angle = 2π − 11π/6 = π/6
→ π/6

Q) θ = −135°
Add 360°: 225° → QIII
Reference angle = 225° − 180° = 45°

R) θ = 17π/6
Subtract 2π = 12π/6: 17π/6 − 12π/6 = 5π/6
5π/6 = 150° → QII
Reference angle = π − 5π/6 = π/6
→ π/6

S) θ = π/3
Already in QI → reference angle = π/3

T) θ = −5π/3
Add 2π: −5π/3 + 6π/3 = π/3 → QI
Reference angle = π/3

Now let’s compile all reference angles clearly:

A) 57°
B) 67°
C) π/6
D) π/3
E) π/3
F) 50°
G) 60°
H) π − 2.3 (or approx 0.84 rad)
I) 80°
J) 1.2
K) 60°
L) 60°
M) π − 2
N) 80°
O) π/3
P) π/6
Q) 45°
R) π/6
S) π/3
T) π/3

But the problem says: “For each of the following, find the reference angle θ′.” It likely expects just the values — possibly as a list.

Since the user didn’t ask for a specific format, and the instruction says: *ONLY THEN provide the final answer*, and the Final Answer section should contain only the answer — but there are 20 parts.

Looking back at the prompt: It says “Solve the problem accurately.” and the image shows a worksheet with A–T.

In such cases, the expected output is a list of reference angles corresponding to A–T.

Given the format constraints, and that ExplainAnswer is false, but we still must do full reasoning before final answer — and the Final Answer should contain only the answers.

I will list them in order, using degrees for degree inputs, radians for radian inputs, simplified exact forms.

Let me double-check each quickly:

A) 57° → 57° ✔
B) 113° → 67° ✔
C) 7π/6 → π/6 ✔
D) 5π/3 → π/3 ✔
E) −2π/3 → coterminal 4π/3 → ref = π/3 ✔
F) −230° → 130° → 50° ✔
G) 300° → 60° ✔
H) 2.3 rad → π − 2.3 ✔ (can’t simplify further)
I) 280° → 80° ✔
J) 1.2 rad → 1.2 ✔ (QI)
K) 420° → 60° ✔
L) −60° → 300° → 60° ✔
M) −2 rad → coterminal = 2π − 2 → QIII → (2π − 2) − π = π − 2 ✔
N) 100° → 80° ✔
O) −4π/3 → 2π/3 → π − 2π/3 = π/3 ✔
P) 11π/6 → π/6 ✔
Q) −135° → 225° → 45° ✔
R) 17π/6 → 5π/6 → π/6 ✔
S) π/3 → π/3 ✔
T) −5π/3 → π/3 ✔

All correct.

Final Answer:
A) 57°
B) 67°
C) π/6
D) π/3
E) π/3
F) 50°
G) 60°
H) π − 2.3
I) 80°
J) 1.2
K) 60°
L) 60°
M) π − 2
N) 80°
O) π/3
P) π/6
Q) 45°
R) π/6
S) π/3
T) π/3