Measuring Angles

Angles in Standard Position

Angles are fundamental in trigonometry and are measured in degrees (°) or radians. An angle in standard position has its vertex at the origin of the coordinate plane, with its initial side along the positive x-axis. The amount of rotation from the initial side to the terminal side determines the angle's measure.

• Initial Side: The starting position of the angle (along the positive x-axis).

• Terminal Side: The position after rotation.

• Positive Angles: Measured counterclockwise from the initial side.

• Negative Angles: Measured clockwise from the initial side.

Types of Angles:

• Acute Angle:

• Right Angle:

• Obtuse Angle:

• Straight Angle:

Example: Sketching angles such as , , and in standard position involves rotating from the positive x-axis by the specified amount, using the appropriate direction (counterclockwise for positive, clockwise for negative).

Complementary & Supplementary Angles

Introduction to Complementary & Supplementary Angles

Complementary and supplementary angles are pairs of angles with specific sum properties:

• Complementary Angles: Two angles whose measures add up to .

• Supplementary Angles: Two angles whose measures add up to .

Example: The complement of is because . The supplement of $20^\circ$ is because .

Note: Complementary and supplementary angles are always considered positive in standard trigonometric problems.

Solving Problems with Complementary & Supplementary Angles

When solving for unknown angles, use the definitions above. In right triangles, the two non-right angles are always complementary. For algebraic expressions, set up equations based on the sum properties.

• Example: If one angle is , its complement is and its supplement is .

• Example: If and are supplementary, then .

Radian Measure and Converting Between Degrees & Radians

Converting Between Degrees & Radians

Radians are another unit for measuring angles, based on the arc length of a circle. One full revolution (circle) is or radians. The conversion formulas are:

• From degrees to radians:

• From radians to degrees:

Example: To convert to radians: radians. To convert radians to degrees: .