Lesson 1: Degree Measure and Classifying Angles
Estimated time: 25-30 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Define an angle and identify its components (vertex, initial side, terminal side)
- Classify angles as acute, right, obtuse, straight, or reflex
- Draw angles in standard position
- Find coterminal angles by adding or subtracting 360 degrees
- Identify complementary and supplementary angle pairs
What Is an Angle?
An angle is formed when two rays share a common endpoint. In trigonometry we think of an angle as a rotation from one ray to another.
Angle — A figure formed by rotating a ray about its endpoint (the vertex). The starting position is the initial side and the ending position is the terminal side.
When the rotation is counterclockwise the angle is positive. When the rotation is clockwise the angle is negative.
Example 1: Positive vs. Negative Angles
A 90° angle is formed by rotating the initial side counterclockwise one-quarter turn.
A −90° angle is formed by rotating the initial side clockwise one-quarter turn.
Both angles look like a right angle, but they have different signed measures because the direction of rotation differs.
Degree Measure
The most familiar unit for measuring angles is the degree. One full rotation equals 360°.
Degree (°) — A unit of angle measurement. One degree is 1/360 of a full rotation. Degrees can be subdivided into minutes (1° = 60′) and seconds (1′ = 60″).
Key benchmark angles to remember:
- 90° – quarter turn (right angle)
- 180° – half turn (straight angle)
- 270° – three-quarter turn
- 360° – full rotation
Example 2: Degrees-Minutes-Seconds to Decimal Degrees
Convert 47° 15′ 36″ to decimal degrees.
Solution:
Step 1: Convert seconds to a fraction of a minute: 36″ ÷ 60 = 0.6′
Step 2: Add to minutes: 15′ + 0.6′ = 15.6′
Step 3: Convert minutes to a fraction of a degree: 15.6′ ÷ 60 = 0.26°
Step 4: Add to degrees: 47° + 0.26° = 47.26°
Classifying Angles by Measure
Angles are classified according to their degree measure:
| Type | Measure | Description |
|---|---|---|
| Acute | 0° < θ < 90° | Less than a right angle |
| Right | θ = 90° | Exactly a quarter turn |
| Obtuse | 90° < θ < 180° | Between a right and straight angle |
| Straight | θ = 180° | A half rotation (a line) |
| Reflex | 180° < θ < 360° | More than a straight angle |
Example 3: Classifying Angles
Classify each angle: (a) 72° (b) 135° (c) 200° (d) 90° (e) 180°
Solution:
(a) 72° is between 0° and 90°, so it is acute.
(b) 135° is between 90° and 180°, so it is obtuse.
(c) 200° is between 180° and 360°, so it is reflex.
(d) 90° is exactly a quarter turn, so it is a right angle.
(e) 180° is exactly a half turn, so it is a straight angle.
Standard Position
In trigonometry we place angles in a coordinate system using a convention called standard position.
Standard Position — An angle is in standard position when its vertex is at the origin and its initial side lies along the positive x-axis. The terminal side is wherever the rotation ends.
From standard position we can describe which quadrant the terminal side falls in:
- Quadrant I: 0° < θ < 90°
- Quadrant II: 90° < θ < 180°
- Quadrant III: 180° < θ < 270°
- Quadrant IV: 270° < θ < 360°
Angles whose terminal side lies exactly on an axis (0°, 90°, 180°, 270°) are called quadrantal angles.
Example 4: Identifying the Quadrant
In which quadrant does the terminal side of 210° lie?
Solution: Since 180° < 210° < 270°, the terminal side is in Quadrant III.
Coterminal Angles
Two angles are coterminal if they share the same terminal side when drawn in standard position. You can find coterminal angles by adding or subtracting full rotations (360°).
Coterminal Angles — Angles that have the same terminal side. If θ is an angle, then θ + 360°n (where n is any integer) is coterminal with θ.
Example 5: Finding Coterminal Angles
Find one positive and one negative angle coterminal with 120°.
Solution:
Positive: 120° + 360° = 480°
Negative: 120° − 360° = −240°
Example 6: Finding the Smallest Positive Coterminal Angle
Find the smallest positive angle coterminal with −150°.
Solution:
−150° + 360° = 210°
Since 210° is between 0° and 360°, it is the smallest positive coterminal angle.
Complementary and Supplementary Angles
Complementary Angles — Two positive angles whose measures sum to 90°. Each angle is called the complement of the other.
Supplementary Angles — Two positive angles whose measures sum to 180°. Each angle is called the supplement of the other.
Example 7: Finding Complements and Supplements
Find the complement and supplement of 35°.
Solution:
Complement: 90° − 35° = 55°
Supplement: 180° − 35° = 145°
Example 8: When a Complement Does Not Exist
Find the complement of 120°.
Solution:
90° − 120° = −30°. Since complements must be positive, 120° has no complement.
Applications: Clocks and Navigation
Degree measure appears everywhere in daily life. Two classic applications involve clocks and compass bearings.
Example 9: Clock Hands
What angle does the minute hand of a clock sweep in 20 minutes?
Solution:
The minute hand completes one full rotation (360°) in 60 minutes.
Rate = 360° / 60 min = 6° per minute
In 20 minutes: 6° × 20 = 120°
Check Your Understanding
Try these questions to see if you have grasped the key concepts:
1. Classify a 175° angle.
2. Find one positive and one negative angle coterminal with 45°.
3. Find the complement and supplement of 62°.
4. In which quadrant does the terminal side of 315° lie?
Key Takeaways
- An angle is a rotation from an initial side to a terminal side; counterclockwise is positive, clockwise is negative.
- One full rotation = 360°. Angles are classified as acute, right, obtuse, straight, or reflex.
- Standard position places the vertex at the origin with the initial side on the positive x-axis.
- Coterminal angles share the same terminal side: add or subtract 360° to find them.
- Complementary angles sum to 90°; supplementary angles sum to 180°.
Ready for More?
Next Lesson
In Lesson 2 you will learn about radian measure and how to convert between degrees and radians.
Start Lesson 2