Coterminal Angles
Reference Angles
Identities
Graphing
Real-World Trig
100
Angle:35°
Find:
- 1 Negative 1 Positive Coterminal Angle (First you get)
Positive Coterminal: 35° + 360°=395°
Negative Coterminal: 35-360= -325°
100
Find the reference angle for:
Angle: 135°
Since 135° is in Quadrant II,
Reference Angle = 180° – 135° = 45°
100
simplify (tanx/secx)
( tanx/secx) = sinx
=(sinx/cosx) / (1/cosx)
= (sinx/cosx) * ( cosx/1)
the cosx cancel each other out
= sinx
100
What is the amplitude of this function?
y=sin(-2x+π)
Amplitude: 1
100
You’re standing 50 feet away from a tall tree. You look up at the top of the tree at an angle of elevation of 48°. Your eyes are 5.5 feet above the ground. How tall is the tree (ft)?
Formula: tan(θ)= opposite/adjacent
tan(48∘) = height above eyes / 50
height above eyes = 50 ⋅ tan(48∘)
≈ 50 ⋅ 1.1106 = 55.53 feet
Total height = 55.5 + 5.5 = 61.0 feet
The tree is about 61.0 feet tall.
200
Angle: -20°
Find:
- 1 Negative 1 Positive Coterminal Angle
Positive Coterminal: -20° + 360° = 340°
Negative Coterminal: -20° - 360° = -380°
200
Find the reference angle for:
Angle:5π/6
Quadrant II,
Reference Angle =π- 5π/6= π/6
200
simplify ( sec^2 x-1)/tanx
( sec^2 x-1)/tanx= tanx
(tan^2x)/tanx
= tanx
200
What is the amplitude of this function?
y=2tan(x-π/4)
Amplitude: none
200
A 15-foot ladder is leaning against a wall. It makes an angle of 60° with the ground. How high up the wall does the ladder reach (ft)?
sin(θ)= opposite/hypotenuse
θ = 60∘
Opposite = height up the wall
Hypotenuse = 15 ft (ladder)
sin(60∘)= height / 15
height = 15 ⋅ sin(60∘) ≈ 15⋅0.8660 = 13.0 feet
The ladder reaches about 13.0 feet up the wall.
300
Angle: 105°
Find:
- 1 Negative 1 Positive Coterminal Angle
Positive Coterminal: 105° + 360° = 465°
Negative Coterminal: 105° − 360° = −255°
300
Find the reference angle for:
Angle: 210°
Quadrant III,
Reference Angle = 210° – 180° = 30°
300
true or false (sinx/1+cosx) + (sinx/1-cosx)=2
false
(sinx/1+cosx) + (sinx/1-cosx)= 2/sinx
(sinx(1+cosx) + sinx(1-cosx))/ 1-cos^2x)
cos cancel out left with 1+1=2
(1-cos^2x) convert to sin^2x
= sinx(2)/sin^2x)
sinx cancel out
= 2/sinx
300
What is the amplitude, period, phase shift, vertical shift of this function?
y=-1/2cos(2x+π/3)
Amplitude: |-1/2|=1/2
Period:2π/2=π
Phase Shift: -π/6
Vertical shift: none
300
An airplane is descending toward a runway. From a point on the ground, then angles of elevation to the airplane is 20o and the plane is flying at an altitude of 1,200 meters. How far is the airplane from the point on the ground?
opposite= 1200m
hypotenuse = unknown
sin(20o)= 1200/x
X=1200m/sin(20o)=1200/0.3420=35,094m
400
Angle: -750°
Find:
- 1 Negative 1 Positive Coterminal Angle
Positive Coterminal: -750+1080=330°
Negative Coterminal: −750° − 360° = −1110°
400
Find the reference angle for:
Angle: 7π/4
This is in Quadrant IV,
Reference Angle = 2π-7π/4= π/4
400
simplify (1+tan^2x)/ sec^2x
(1+tan^2x)/ sec^2x=1
(1+tan^2x) convert to sec^2x
(sec^2x/sec^2x)=1
400
Graph this function and label the key points.
y=cos(x-π/2)
400
A radio station tower was built in two sections. From a point 100 ft from the base of the tower, the angle of elevation of the top of the first section is , 30º. the angle of elevation of the top of the second section is 40º. What is the height of the top section of the tower?
tan(30º)= a/100
0.57735=a/100
*mulitply 100 to the another side
a=57.73 ft
tan(40º)= b/100
0.83909=b/100
*multiply 100 to the another side
b=83.90 ft
(b-a)=83.90-57.73=26.17 ft
500
Angle: -246,195°
Find:
- 1 Negative 1 Positive Coterminal Angle (First you get)
Positive Coterminal: -246,195+246,240 =45°
Negative Coterminal: -246,195-360=-246,555
500
Find reference angle of -1073π/6
-1073π/6+ 90* 12π/6= -1073π/6 + 1080π/6 = 7π/6
Reference Angle= 7π/6 - π = 7π/6- 6π/6= π/6
500
true or false is ( tanx)/ (1+tan^2x)=sinx * cosx
true
( tanx)/ (1+tan^2x)=sinx * cosx
tan x converts to (sinx/cosx) and 1+tan^2x converts to secx to then you can change to (1/cos^2x)
=(sinx/cosx)/(1/cos^2x)
= (sinx/cosx) * (cos^2x)
two cosx will cancels out levae you with just one cosx = sinx * cosx
500
What is the amplitude, period, phase shift, and vertical shift of this function? Graph the function as well and you must label the 5 key points. -2sin(3x+π/2)+1
Amplitude: |-2|=2
Period:2π/3
Phase shift: -π/6 left
Vertical shift: up 1
500
Two people are on different buildings, looking up at the same airplane flying in a straight line. The first person sees the airplane at an angle of elevation of 40°, and the second person, at the next building sees the other side of the airplane, at an angle of elevation of 50°. The airplane is flying at a constant height of 1,000 meters. Which person is closer to the airplane?
Distance= opposite/ sin(x)
First person(40o):
d1= 1000/sin(40o)= 1000/0.6428 = 1,556.2m
Second person (50o ):
d1 = 1000/ sin(50o)= 1000/0.7660 =1,305.6m
The person that is closer is: Person #2