Finding Exact Values of Trig Ratios

Special Right Triangles

• 30-60-90

Special Right Triangles

• 45-45-90

Consider the Special Angles in relation to the Unit Circle…

• Since the radius is always 1 and we know that the sine, cosine, and tangent values are always determined by the point through which the terminal ray passes, we can construct the following….

30 Degree Angle

30sin 30cos 30tan

60 Degree Angle

60sin 60cos 60tan

45 Degree Angle

45sin 45cos 45tan

Values to Memorize

60 tan 45tan 30tan

60cos 45cos 30cos

sin60 45sin 30sin

Reference Angles

• We have previously learned that if the angle is the same size, then it will have the same values (but the + or – may switch depending upon the quadrant)

• A Reference Angle is the acute angle formed by the terminal ray and the x-axis– The Reference Angle tells us the size of the triangle

Examples

• Give the reference angle for each of the following angles:

315

Examples

• Give the reference angle for each of the following angles:

315 240

Examples

• Give the reference angle for each of the following angles:

315 240

6

5

Examples

• Give the reference angle for each of the following angles:

315 240

6

54

7

Reference Angles (in Radians)

• What can we observe about the reference angle in each radian measure?

Exact Values of Special Angles

• Find the exact value for each of the following:120sin

Exact Values of Special Angles

• Find the exact value for each of the following:120sin 150cos

Exact Values of Special Angles

• Find the exact value for each of the following:120sin 150cos 225tan

Exact Values of Special Angles

• Find the exact value for each of the following:120sin 150cos 225tan

6

7sin

Exact Values of Special Angles

• Find the exact value for each of the following:120sin 150cos 225tan

6

7sin

4

5cos

Exact Values of Special Angles

• Find the exact value for each of the following:120sin 150cos 225tan

6

7sin

4

5cos

3

4tan

Homework

• Pg. 280 (1-4, 11-18)