Try It
3π 2 3π 2
−135° −135°
7π 10 7π 10
α=150° α=150°
β=60° β=60°
7π 6 7π 6
215π 18 =37.525units 215π 18 =37.525units
1.88
3π 2 3π 2 rad/s
1655 kilometers per hour
7 25 7 25
sin t = 33 65 , cos t = 56 65 , tan t = 33 56 , sec t = 65 56 , csc t = 65 33 , cot t = 56 33 sin t = 33 65 , cos t = 56 65 , tan t = 33 56 , sec t = 65 56 , csc t = 65 33 , cot t = 56 33
sin( π 4 ) = 2 2 ,cos( π 4 )= 2 2 ,tan( π 4 )=1, sec( π 4 ) = 2 ,csc( π 4 )= 2 ,cot( π 4 )=1 sin( π 4 ) = 2 2 ,cos( π 4 )= 2 2 ,tan( π 4 )=1, sec( π 4 ) = 2 ,csc( π 4 )= 2 ,cot( π 4 )=1
2
adjacent=10;opposite=10 3 ; adjacent=10;opposite=10 3 ; missing angle is π 6 π 6
About 52 ft
cos(t)=− 2 2 ,sin(t)= 2 2 cos(t)=− 2 2 ,sin(t)= 2 2
cos(π)=−1,sin(π)=0 cos(π)=−1,sin(π)=0
sin(t)=− 7 25 sin(t)=− 7 25
approximately 0.866025403
π 3 π 3
ⓐ cos(315°)= 2 2 ,sin(315°)= – 2 2 cos(315°)= 2 2 ,sin(315°)= – 2 2
ⓑ cos( − π 6 )= 3 2 ,sin( − π 6 )=− 1 2 cos( − π 6 )= 3 2 ,sin( − π 6 )=− 1 2
( 1 2 ,− 3 2 ) ( 1 2 ,− 3 2 )
sint=− 2 2 cost= 2 2 ,tant=−1,sect= 2 ,csct=− 2 ,cott=−1 sint=− 2 2 cost= 2 2 ,tant=−1,sect= 2 ,csct=− 2 ,cott=−1
sinπ3=32,cosπ3=12,tanπ3=3,secπ3=2,cscπ3=233,cotπ3=33 sinπ3=32,cosπ3=12,tanπ3=3,secπ3=2,cscπ3=233,cotπ3=33
sin( −7π 4 )= 2 2 ,cos( −7π 4 )= 2 2 ,tan( −7π 4 )=1, sec( −7π 4 )= 2 ,csc( −7π 4 )= 2 ,cot( −7π 4 )=1 sin( −7π 4 )= 2 2 ,cos( −7π 4 )= 2 2 ,tan( −7π 4 )=1, sec( −7π 4 )= 2 ,csc( −7π 4 )= 2 ,cot( −7π 4 )=1
− 3 − 3
−2 −2
sint sint
cost=− 8 17 ,sint= 15 17 ,tant=− 15 8 csct= 17 15 ,cott=− 8 15 cost=− 8 17 ,sint= 15 17 ,tant=− 15 8 csct= 17 15 ,cott=− 8 15
sint=−1,cost=0,tant=Undefined sect=Undefined,csct=−1,cott=0 sint=−1,cost=0,tant=Undefined sect=Undefined,csct=−1,cott=0
sect= 2 ,csct= 2 ,tant=1,cott=1 sect= 2 ,csct= 2 ,tant=1,cott=1
≈−2.414 ≈−2.414
7.1 Section Exercises
Whether the angle is positive or negative determines the direction. A positive angle is drawn in the counterclockwise direction, and a negative angle is drawn in the clockwise direction.
Linear speed is a measurement found by calculating distance of an arc compared to time. Angular speed is a measurement found by calculating the angle of an arc compared to time.
240° 240°
4π 3 4π 3
2π 3 2π 3
7π 2 ≈11.00 in 2 7π 2 ≈11.00 in 2
81π 20 ≈12.72 cm 2 81π 20 ≈12.72 cm 2
20° 20°
60° 60°
−75° −75°
π 2 π 2 radians
−3π −3π radians
ਪਾਈ ਰੇਡੀਅਨ
ਪੰਜ ਪਾਈ ਬਟਾ ਛੇ ਰੇਡੀਅਨ
5.02π 3 ≈5.26 ਮੀਲ
25π 9 ≈8.73 ਸੈਂਟੀਮੀਟਰ
21π 10 ≈6.60 ਮੀਟਰ
104.7198 ਵਰਗ ਸੈਂਟੀਮੀਟਰ
0.7697 ਵਰਗ ਇੰਚ
250°
320°
4π ਬਟਾ 3
8π ਬਟਾ 9
1320 ਰੇਡੀਅਨ ਪ੍ਰਤੀ ਮਿੰਟ = 210.085 ਪ੍ਰਤੀ ਮਿੰਟ ਚੱਕਰ
7 ਇੰਚ ਪ੍ਰਤੀ ਸੈਕੰਡ, 4.77 ਪ੍ਰਤੀ ਮਿੰਟ ਚੱਕਰ, 28.65 ਡਿਗਰੀ ਪ੍ਰਤੀ ਸੈਕੰਡ
1,809,557.37 ਮਿਲੀਮੀਟਰ ਪ੍ਰਤੀ ਮਿੰਟ = 30.16 ਮੀਟਰ ਪ੍ਰਤੀ ਸੈਕੰਡ
5.76 ਮੀਲ
120°
794 ਮੀਲ ਪ੍ਰਤੀ ਘੰਟਾ
2,234 ਮੀਲ ਪ੍ਰਤੀ ਘੰਟਾ
11.5 ਇੰਚ
7.2 ਭਾਗ ਅਭਿਆਸ
ਕਿਸੇ ਕੋਣ ਦਾ ਟੈਨਜੈਂਟ (ਸਪਰਸ਼ ਰੇਖਾ) ਉਸਦੇ ਸਾਹਮਣੇ ਵਾਲੀ ਭੁਜਾ ਅਤੇ ਨਾਲ ਲੱਗਦੀ ਭੁਜਾ ਦਾ ਅਨੁਪਾਤ ਹੁੰਦਾ ਹੈ।
ਉਦਾਹਰਨ ਵਜੋਂ, ਕਿਸੇ ਕੋਣ ਦਾ ਸਾਈਨ ਉਸਦੇ ਪੂਰਕ ਕੋਣ ਦੇ ਕੋਸਾਈਨ ਦੇ ਬਰਾਬਰ ਹੁੰਦਾ ਹੈ; ਕਿਸੇ ਕੋਣ ਦਾ ਕੋਸਾਈਨ ਉਸਦੇ ਪੂਰਕ ਕੋਣ ਦੇ ਸਾਈਨ ਦੇ ਬਰਾਬਰ ਹੁੰਦਾ ਹੈ।
ਪਾਈ ਬਟਾ ਛੇ
ਪਾਈ ਬਟਾ ਚਾਰ
b= 20 3 3 ,c= 40 3 3 b= 20 3 3 ,c= 40 3 3
a=10,000,c=10,000.5 a=10,000,c=10,000.5
b= 5 3 3 ,c= 10 3 3 b= 5 3 3 ,c= 10 3 3
5 29 29 5 29 29
5 2 5 2
29 2 29 2
5 41 41 5 41 41
5 4 5 4
41 4 41 4
c=14,b=7 3 c=14,b=7 3
a=15,b=15 a=15,b=15
b=9.9970,c=12.2041 b=9.9970,c=12.2041
a=2.0838,b=11.8177 a=2.0838,b=11.8177
a=55.9808,c=57.9555 a=55.9808,c=57.9555
a=46.6790,b=17.9184 a=46.6790,b=17.9184
a=16.4662,c=16.8341 a=16.4662,c=16.8341
188.3159
200.6737
498.3471 ft
1060.09 ft
27.372 ft
22.6506 ft
368.7633 ft
7.3 Section Exercises
The unit circle is a circle of radius 1 centered at the origin.
Coterminal angles are angles that share the same terminal side. A reference angle is the size of the smallest acute angle, t, t, formed by the terminal side of the angle t t and the horizontal axis.
The sine values are equal.
I
IV
3 2 3 2
1 2 1 2
2 2 2 2
0
-1
3 2 3 2
60° 60°
80° 80°
45° 45°
π 3 π 3
π 3 π 3
π 8 π 8
60°, 60°, Quadrant IV, sin(300°)=− 3 2 sin(300°)=− 3 2,cos(300°)= 1 2 cos(300°)= 1 2
45°, 45°, Quadrant II, sin( 135° )= 2 2 sin( 135° )= 2 2,cos(135°)=− 2 2 cos(135°)=− 2 2
60°, 60°, Quadrant II, sin( 120° )= 3 2 sin( 120° )= 3 2,cos(120°)=− 1 2 cos(120°)=− 1 2
30°, 30°, Quadrant II, sin( 150° )= 1 2 sin( 150° )= 1 2,cos(150°)=− 3 2 cos(150°)=− 3 2
π 6 , π 6 , Quadrant III, sin( 7π 6 )=− 1 2 sin( 7π 6 )=− 1 2,cos( 7π 6 )=− 3 2 cos( 7π 6 )=− 3 2
π 4 , π 4 , Quadrant II, sin( 3π 4 )= 2 2 sin( 3π 4 )= 2 2,cos( 3π 4 )=− 2 2 cos( 3π 4 )=− 2 2
π 3 , π 3 , Quadrant II, sin( 2π 3 )= 3 2 sin( 2π 3 )= 3 2,cos( 2π 3 )=− 1 2 cos( 2π 3 )=− 1 2
π 4 , π 4 , Quadrant IV, sin( 7π 4 )=− 2 2 ,cos( 7π 4 )= 2 2 sin( 7π 4 )=− 2 2 ,cos( 7π 4 )= 2 2
77 9 77 9
− 15 4 − 15 4
( −10,10 3 ) ( −10,10 3 )
( –2.778, 15.757 ) ( –2.778, 15.757 )
[ –1, 1 ] [ –1, 1 ]
sint= 1 2 ,cost=− 3 2 sint= 1 2 ,cost=− 3 2
sint=− 2 2 ,cost=− 2 2 sint=− 2 2 ,cost=− 2 2
sint= 3 2 ,cost=− 1 2 sint= 3 2 ,cost=− 1 2
sint=− 2 2 ,cost= 2 2 sint=− 2 2 ,cost= 2 2
sint=0,cost=−1 sint=0,cost=−1
sint=−0.596,cost=0.803 sint=−0.596,cost=0.803
sint= 1 2 ,cost= 3 2 sint= 1 2 ,cost= 3 2
sint=− 1 2 ,cost= 3 2 sint=− 1 2 ,cost= 3 2
sint=0.761,cost=−0.649 sint=0.761,cost=−0.649
sint=1,cost=0 sint=1,cost=0
−0.1736
0.9511
−0.7071
−0.1392
−0.7660
2 4 2 4
− 6 4 − 6 4
2 4 2 4
2 4 2 4
0
( 0,–1 ) ( 0,–1 )
37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds
7.4 Section Exercises
Yes, when the reference angle is π 4 π 4 and the terminal side of the angle is in quadrants I and III. Thus, a x= π 4 , 5π 4 , x= π 4 , 5π 4 , the sine and cosine values are equal.
Substitute the sine of the angle in for y y in the Pythagorean Theorem x 2 + y 2 =1. x 2 + y 2 =1. Solve for x x and take the negative solution.
The outputs of tangent and cotangent will repeat every π π units.
2 3 3 2 3 3
3 3
2 2
1
2
3 3 3 3
− 2 3 3 − 2 3 3
3 3
− 2 − 2
–1
-2
− 3 3 − 3 3
2
3 3 3 3
–2
–1
sint=− 2 2 3 sint=− 2 2 3, sect=−3sect=−3, csct=− 3 2 4 csct=− 3 2 4, tant=2 2 tant=2 2, cott= 2 4 cott= 2 4
sect=2, sect=2, csct= 2 3 3 , csct= 2 3 3 , tant= 3 , tant= 3 , cott= 3 3 cott= 3 3
− 2 2 − 2 2
3.1
1.4
sint= 2 2 sint= 2 2, cost= 2 2 cost= 2 2, tant=1tant=1, cott=1cott=1, sect= 2 sect= 2, csct= 2 csct= 2
sint=− 3 2 sint=− 3 2, cost=− 1 2 cost=− 1 2tant= 3 tant= 3, cott= 3 3 cott= 3 3, sect=−2sect=−2, csct=− 2 3 3 csct=− 2 3 3
–0.228
–2.414
1.414
1.540
1.556
sin( t )≈0.79 sin( t )≈0.79
csct≈1.16 csct≈1.16
even
even
sint cost =tant sint cost =tant
13.77 hours, period: 1000π 1000π
3.46 inches
Review Exercises
45° 45°
− 7π 6 − 7π 6
10.385 meters
60° 60°
2π 11 2π 11
1036.73 miles per hour
2 2 2 2
3 3 3 3
72° 72°
a= 10 3 ,c= 2 106 3 a= 10 3 ,c= 2 106 3
6 11 6 11
a= 5 3 2 ,b= 5 2 a= 5 3 2 ,b= 5 2
369.2136 ft
2 2 2 2
60° 60°
3 2 3 2
all real numbers
3 2 3 2
2 3 3 2 3 3
2
–2.5
1 3 1 3
cosine, secant
Practice Test
150° 150°
6.283 centimeters
15° 15°
3.351 feet per second, 2π 75 2π 75 radians per second
a= 9 2 ,b= 9 3 2 a= 9 2 ,b= 9 3 2
1 2 1 2
real numbers
1
− 2 − 2
–0.68
π 3 π 3