Trigonometric Functions of Any Angles
Objective:
To evaluate trig functions of any angle by using reference angles
θ(x, y)
r
2 2r = x + y
sin θ =yr
cos θ =xr
tan θ =yx
csc θ =ry
sec θ =rx
cot θ =xy
Evaluate the six trigonometric functions if the point (-4,3) lies on the terminal side of an angle θ.
2 2r = -4 + 3
r = 16 + 9
r = 25
r = 5
sin θ =35
cos θ =-45
tan θ =3-4
csc θ =53
sec θ =5-4
cot θ =-43
Evaluate the six trigonometric functions if the point (1,-3) lies on the terminal side of an angle θ.
2 2r = 1 + -3
r = 1 + 9
r = 10
sin θ =
cos θ =
tan θ =
csc θ =
sec θ =
cot θ =
-3
10-3 10
10
1
101010
-31
-3
10-3
101
1-3
Evaluate the six trigonometric functions if the point (-7,-4) lies on the terminal side of an angle θ.
2 2r = -7 + -4
r = 49+16
sin θ =
cos θ =
tan θ =
csc θ =
sec θ =
cot θ =
-4
65-4 65
65
-7
65-7 65
65
-4-7
47
65-4
65-7
74
r = 65
Evaluate the six trigonometric functions if the point (3, 0) lies on the terminal side of an angle θ.
2 2r = 3 + 0
r = 9
r = 3
sin θ =
cos θ =
tan θ =
csc θ =
sec θ =
cot θ =
03
0
33
1
03
0
30
= undefined
130
= undefined
When Trig Functions are +/-
(+,+)(-,+)
(-,-) (+,-)
sin, csc +
cos, sec –
tan, cot –
sin, csc –
cos, sec –
tan, cot +
sin, csc –
cos, sec +
tan, cot –
sin, csc +
cos, sec +
tan, cot +III
III IV
Name the quadrant in which the angle lies.
sinθ < 0 and cosθ < 0
sinθ > 0 and tanθ < 0
III
II
Evaluate the six trigonometric functions if the point (0, 3) lies on the terminal side of an angle θ.
2 2r = 0 + 3
r = 9
r = 3
sin θ =
cos θ =
tan θ =
csc θ =
sec θ =
cot θ =
03
0
33
1
03
30
= undefined
1
30
= undefined 0
Evaluate the six trigonometric functions if the point (-3, 0) lies on the terminal side of an angle θ.
2 2r = -3 + 0
r = 9
r = 3
sin θ =
cos θ =
tan θ =
csc θ =
sec θ =
cot θ =
03
0
-33
1
0-3
0
30
= undefined
-1-30
= undefined
Evaluate the six trigonometric functions if the point (0, -3) lies on the terminal side of an angle θ.
2 2r = 0 + -3
r = 9
r = 3
sin θ =
cos θ =
tan θ =
csc θ =
sec θ =
cot θ =
-1
03
0
0-3
0
-33
= -1
30
-30
= undefined
= undefined
(3,0)
If (3,0) is on the terminal side of the angle, then θ = 0°.
(0,3)
If (0,3) is on the terminal side of the angle, then θ = 90°
90°
(-3,0)
If (-3,0) is on the terminal side of the angle, then θ = 180°
(0,-3)
If (0,-3) is on the terminal side of the angle, then θ = 270°
180°270°
Quadrantal Angle
• Angle whose terminal side falls on an axis.
• Examples: 0°, 90°, 180°, 270°, 360°
Trig Functions of Quadrantal Angles
θ sin cos tan csc sec cot
0°
90°
180°
270°
360°
0 1 0 ud. 1 ud.
1 0 ud. 1 ud. 0
0 -1 0 ud. -1 ud.
-1 0 ud. -1 ud. 0
0 1 0 ud. 1 ud.
Reference Angles
• The acute angle formed by the terminal side of the angle and the horizontal axis (x-axis).
Remember:Must be positiveMust be acute
Find the reference angle.
140°
180° – 140° = 40°
Find the reference angle.
230°
230° – 180°
= 50°
Short-Cuts for Reference Angles
Quadrant θ in degrees θ in radians
I
II
III
IV
If θ goes around more than once…
θ θ
180° - θ π - θ
θ - 180° θ - π
360° - θ 2π - θθ - 360°.
Then proceed with above.
θ - 2π
Then proceed with above.
Find the reference angle.
310
170
305
360° - 310° = 50°
180° - 170 ° = 10°
360° - 305° = 55°
Find the reference angle.
3
2
5
7
4
2
3
2
1 3
3 2
3 3
3
5
72
4
2 7
1 4
8 7
4 4
4
Evaluate the following.
cos 225
tan120
225° - 180° = 45°
cos(45°)22
180° - 120° = 60°
tan(60°) 3
22
3
Quadrant III
Quadrant II
In Quadrant III, cos is negative.
In Quadrant II, tan is negative.
Evaluate the following.
tan 225 225 180 45
tan 45 1Quadrant III
1
In Quadrant III, tan is positive.
Evaluate the following.
sin 210
210 180 30
sin 30 1
2
Quadrant III1
2
In Quadrant III, sin is negative.
= 55°
= 70°
Find the reference angle.
375
470
595
375° - 360° = 15°
470° - 360 ° = 110°
595° - 360° = 235°
180° - 110 °
235° - 180 °
Evaluate the following.
sin 390
390° - 360° = 30°
sin(30°)12
It’s larger than 360°!
Quadrant I.
12
In Quadrant I, sin is positive.
Evaluate the following.
9tan
4
It’s larger than 2π!
92
4
9 2
4 1
9 8
4 4
4
tan4
1
Quadrant I!1
If θ is negative…
1. Add 360° (or 2π) until you get a positive angle.
2. Proceed as usual.
= 68°
Find the reference angle.
-275
-190
-112
-275° + 360°= 85°
-190° + 360 °= 170°
-112° + 360°= 248°
248° - 180 °
180° - 170° = 10°
Find the reference angle.
3
4
32
4
3 2
4 1
3 8
4 4
5
4
Quadrant III.5
4
5
4 1
5 4
4 4
4
Evaluate the following.
sin( 120 )
-120 + 360 = 240°It’s negative.
Quadrant III.
240° - 180° = 60°
sin(60°)32
32
In Quadrant III, sin is negative.