12.4 NOTES Trigonometric Functions of Any Angle
1
BELLWORK #1: Calculate the following trig functions. What do you notice?
A) sin(45°) =
B) sin(135°) =
C) sin(225°) =
D) sin(315°) =
√22
√22√22
√22
-
-
BELLWORK #2: Calculate the following trig functions. What do you notice?
A) cos(60°) =
B) cos(120°) =
C) cos(240°) =
D) cos(300°) =
12
12
-
12
-
12
BELLWORK #3: What Greek letter is this?
θ"theta"
y
θ
(x, y)
r
LESSON 12. 4 - Trig Functions of Any Angle
• Let θ be an angle in standard position and let (x, y) be the point where the terminal side of θ intersects the circle x2 + y2 = r2. • The six trigonometric functions of θ can are defined as follows:
x
sin θ = csc θ =yr
ry y ≠ 0
cos θ = sec θ =xr
rx x ≠ 0
tan θ = cot θ =yx
xy y ≠ 0x ≠ 0 = -0.75
= -1.6667
= 1.25
= -1.3333
= -0.6
Let (‑3, 4) be a point on the coterminal side of angle θ in standard position. Calculate the six trigonometric function of angle θ.
sin θ =
cos θ =
tan θ =
csc θ =
sec θ =
cot θ =
y
θ
(‑3, 4)r
x
45
= 0.8
-354-3545-3-34
-3
45
42 + (-3)2 = r2
16 + 9 = r2
25 = r2
5 = r
12.4 NOTES Trigonometric Functions of Any Angle
2
= -0.3846
Let (12, ‑5) be a point on the coterminal side of angle θ in standard position. Calculate the six trigonometric function of angle θ.
y
θ
(12, ‑5)r
x
sin θ =
cos θ =
tan θ =
csc θ =
sec θ =
cot θ =
-513
122 + (-5)2 = r2
144 + 25 = r2
169 = r2
13 = r
13-5
12 = 0.92311213
= -0.4167-512
= -2.613-5
= 1.08331312
= -2.412-5
Let (‑8, ‑15) be a point on the coterminal side of angle θ in standard position. Calculate the six trigonometric function of angle θ.
y
θ
(‑8, ‑15)r
x
= -0.8824sin θ =
cos θ =
tan θ =
csc θ =
sec θ =
cot θ =
-1517
(-8)2 + (-15)2 = r2
64 + 225 = r2
289 = r2
17 = r
= 0.4706-817
= 1.875-15-8
= -1.133317
-15
= -2.12517-8
= 0.5333-8
-15
-8
-15= 17
THE UNIT CIRCLE
• The circle x2 + y2 = 1, which has a center at (0, 0) and a radius length of 1, is called the UNIT CIRCLE.
• To simplify calculations, we like to place an angleʹs triangle into the unit circle, because its radius is 1. This means that the trig functions will only depend on what x and y are.
y
θ
(x, y)
r = 1
x
• In the unit circle, sin(θ) = y, cos(θ) = x, and tan(θ) = xy
COMMON ANGLES IN TRIGONOMETRY
• The angles 30°, 45°, and 60° occur frequently in trig. Those angles create special right triangles, whose sides are always in a certain ratio:
30°
60°
45°
45°
1
√3
2 √2 1
1Reduce each hypotenuse to be equal to 1 (and adjust other side
lengths proportionally) to make the Unit Circle ratios below:
Find the value of each trig function.
A) sin(45°) =
B) cos(60°) =
C) tan(30°) =
2212
√33
Find the value of each trig function.
A) cot(45°) =
B) csc(60°) =
C) sec(30°) =
1
2√32
2√32
12.4 NOTES Trigonometric Functions of Any Angle
3
Find the value of each trig function.
G) cos =
H) tan =
I) sin =
π3π6
π4
12
√33
√22
Find the value of each trig function.
G) csc =
H) sec =
I) cot =
π3π6
π4
2√32
2√32
1
REFERENCE ANGLES
• Let θ be an angle in standard position. The
REFERENCE ANGLE for θ is the acute angle θʹ formed by the terminal side of θ and the x‑axis.
• Reference angles are only calculated for angles between 90° and 360°.
• If an angle is less than 90°, then it is already acute.
• If an angle is greater than 360° or less than or equal to 0°, first find its coterminal angle between 90° and 360°, then calculate the reference angle measure.
What is the reference angle?
θ = 100°θθ'
θʹ = 80°
What is the reference angle?
θ = 205°
'θʹ = 25°
What is the reference angle?
θ = 660°
'
θʹ = 60°
12.4 NOTES Trigonometric Functions of Any Angle
4
What is the reference angle?
θ = ‑315°
'θʹ = 45°
What is the reference angle?
θ = ‑570°
'θʹ = 30°
What is the reference angle?
2π3θ =
'
θʹ = π3
What is the reference angle?
7π4
θ =
'θʹ = π4
What is the reference angle?
19π6θ =
'θʹ = π6
What is the reference angle?
7π6
‑θ =
'
θʹ = π6
12.4 NOTES Trigonometric Functions of Any Angle
5
What is the reference angle?
14π3
‑θ =
'θʹ = π3
REFERENCE ANGLES
• Reference angles allow you to find a trigonometric function for any angle θ.
• The sign of the trigonometric function will depend on in which quadrant θ lies.
Evaluate the trigonometric function using reference angles.
sin(495°)495° is coterminal with 135°, which is in Quadrant II.
The reference angle for 135° is 45°.
So, sin(45°) =
The sine function is positive in Quadrant II, so the final answer is
sin(495°) =
√22
√22
Evaluate the trigonometric function using reference angles.
tan(570°)570° is coterminal with 210°, which is in Quadrant III.
The reference angle for 210° is 30°.
So, tan(30°) =
The tangent function is positive in Quadrant III, so the final answer is
tan(570°) =
√33
√33
Evaluate the trigonometric function using reference angles.
cos(‑240°)-240° is coterminal with 120°, which is in Quadrant II.
The reference angle for 120° is 60°.
So, cos(60°) =
The cosine function is negative in Quadrant II, so the final answer is
cos(-240°) = -
12
12
Evaluate the trigonometric function using reference angles.
sin(‑390°)-390° is coterminal with 330°, which is in Quadrant IV.
The reference angle for 330° is 30°.
So, sin(30°) =
The sine function is negative in Quadrant IV, so the final answer is
sin(-390°) = -
12
12
12.4 NOTES Trigonometric Functions of Any Angle
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Evaluate the trigonometric function using reference angles.
8π3cos
is coterminal with , which is in Quadrant II.
The reference angle for is .
So, cos( ) =
The cosine function is negative in Quadrant II, so the final answer is
cos( ) = -
12
12
8π3
2π3
2π3
π3
π3
8π3
Evaluate the trigonometric function using reference angles.
15π4sin
is coterminal with , which is in Quadrant IV.
The reference angle for is .
So, sin( ) =
The sine function is negative in Quadrant IV, so the final answer is
sin( ) = -
15π4
7π4
7π4
π4
π4
22
√22
15π4
Evaluate the trigonometric function using reference angles.
‑5π6tan
is coterminal with , which is in Quadrant III.
The reference angle for is .
So, tan( ) =
The tangent function is positive in Quadrant III, so the final answer is
tan( ) =
-5π6
7π6
7π6
π6
π6
√33
-5π6
√33
Evaluate the trigonometric function using reference angles.
‑11π3
cos
is coterminal with , which is in Quadrant I.
Since is already in Quadrant I, there is not need to find a reference angle.
So, cos( ) =
The cosine function is positive in Quadrant I, so the final answer is
cos( ) =
12
12
-11π3
π3
π3
π3
-11π3
HOMEWORK:12.4 Worksheet ‑ Trig Functions of Any Angle