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Section 6.3Properties of the
Trigonometric Functions
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If sin > 0 and cos < 0, name the quadrant in which the angle lies.
For sin > 0 the y value must be positive so the angle must be in quadrant I or II.
For cos < 0 the x value must be negative so the angle must be in quadrant II or III.
Therefore, this angle must lie in quadrant II.
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10 3 10Given sin and cos , find the value of each of the four remaining 10 10
trigonometric functions of .
10sin 10 10 110tancos 10 33 10 3 10
10
1 1cot 31tan
3
1 1 10csc 10sin 1 1
100 0
1 1 10 10sec
cos 33 10 3 1010
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22
Find the exact value of each expression. Do not use a calculator.
cos1 3(a) cos 35 (b) cotcsc 35 3sin
3
2 2 22
1(a) cos 35 sin 35 cos 35 1csc 35
cos3(b) cot cot cot 0
3 3 3sin3
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2Given that sin and cos , find the exact value of each 5
of the remaining five trigonometric functions of .
21cos = 5
xr
5csc = 2
ry
5 5 21sec = 2121
rx
21cot = 2
xy
θP(x,2)
r = 5
2 2 21tan = 2121
yx
2Since sin 0 and cos 0, is in quadrant II.5
2 2 2The circle has a radius of 5 so its equation is 5x y
2 2 2 is a point on the circle so 5 2P x
21x 2 21x
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2Given that sin and is an acute angle, find the exact value of each 5
of the remaining five trigonometric functions of .
2 2sin c 1os 2
22 co 1s5
2 4 2112 2
cos5 5
21 212 5
cos5
Since θ is in quadrant II, x values are negative
2sin 2 55tancos 521 21
5
2 2121
1 21cottan 2
1 1 5csc
2sin 25
1 1 5 5 21seccos 2121 21
5
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P(-1,-3)
θ
1Given that cot and sin , find the exact value of each 3
of the remaining five trigonometric functions of .
1 10cos = 1010
xr
3tan = 31
yx
10csc = 3
ry
10sec = 101
rx
3 3 10sin = 1010
yr
1Since cot 0 and sin 0, is in quadrant III.3
2 2 2The circle has a radius of 3 so its equation is 3x y
2 is a point on the circle so 1 9 10P r
10r
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2 2cot 1 csc 2
21 1 csc3
2 1csc9
1019
10 109 3
csc
Since sin θ < 0, csc θ is negative.
sin 3 10 110costan 3
3 1010
110 3 0
1 1tan 31cot3
1 1 3 3 10sincsc 1010 10
3
1 1 10sec 10cos 10 1
100
1Given that cot and sin , find the exact value of each 3
of the remaining five trigonometric functions of .
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Find the exact value of:
(a) cos (60°) (b) sin (390°) (c) tan 374
1(a) cos 60 cos602
1(b) sin 390 sin 390 sin 30 360 sin 302
37 37 36(c) tan tan tan4 4 4 4
tan 9 tan 14 4