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EXAMPLE 1
Evaluate trigonometric functions given a point
Let (–4, 3) be a point on the terminal side of an angle θ
in standard position. Evaluate the six trigonometric
functions of θ.
SOLUTION
Use the Pythagorean theorem to find the value of r.
r = √ x2 + y2
= √ (–4)2 + 32 = √ 25 = 5
EXAMPLE 1
Evaluate trigonometric functions given a point
Using x = –4, y = 3, and r = 5, you can write the
following:
3
4
y
x
–
sin θ =
cos θ =
= 5
=
5
r
r
tan θ =
y
x
sec θ =
r
x
=–
3
4
=–
5
4
5
3
csc θ =
r
y
=
cot θ =
x
y
4
=–
3
EXAMPLE 2
Use the unit circle
Use the unit circle to evaluate the six trigonometric
functions of θ = 270°.
SOLUTION
Draw the unit circle, then draw the
angle θ = 270° in standard
position. The terminal side of θ
intersects the unit circle at (0, –1),
so use x = 0 and y = –1 to evaluate
the trigonometric functions.
EXAMPLE 2
Use the unit circle
sin θ =
y
r
–1
= 1 = –1
cos θ =
x
r
0
= 1 =0
tan θ =
y
x
–1
= 0 undefined
csc θ =
sec θ =
r
y
r
x
cot θ =
1
= – 1 = –1
1
= 0
x
y
undefined
0
= –1 = 0
GUIDED PRACTICE
for Examples 1 and 2
Evaluate the six trigonometric functions of θ.
1.
SOLUTION
Use the Pythagorean Theorem to find the value of r.
r = √ x2 + y2
= √ 32 + (–3)2 = √ 18 = 3√ 2
GUIDED PRACTICE
for Examples 1 and 2
Using x = 3, y = –3 , and r = 3√ 2, you can write the
following:
3
y
x
√2
–
–
– 3 =
=
sin θ =
cos θ =
=
=
2
r
3√ 2
r
3√ 2
3
3 = –1
tan θ =
y
x
=–
sec θ =
r
x
= 3√ 2 = √ 2
3
√2
2
csc θ =
r
3√ 2
= – 3 = –√ 2
y
cot θ =
x
y
3
=–
= –1
3
GUIDED PRACTICE
for Examples 1 and 2
2.
SOLUTION
Use the Pythagorean theorem to find the value of r.
r = √ (–8)2 + (15)2 = √ 64 + 225 = √ 289 = 17
GUIDED PRACTICE
for Examples 1 and 2
Using x = –8, y = 15, and r = 17, you can write the
following:
sin θ =
y
r
15
= 17
tan θ =
y
x
15
–
=
8
sec θ =
r
x
17
–
=
8
cos θ =
x
r
8
= – 17
csc θ =
r
y
17
= 15
cot θ =
x
y
=–
8
15
GUIDED PRACTICE
for Examples 1 and 2
3.
SOLUTION
Use the Pythagorean theorem to find the value of r.
r = √ x2 + y2
= √ (–5)2 + (–12)2 = √ 25 + 144 = 13
GUIDED PRACTICE
for Examples 1 and 2
Using x = –5, y = –12, and r = 13, you can write the
following:
y
x
5
12
–
–
sin θ =
cos θ =
=
=
13
13
r
r
tan θ =
y
x
= 12
5
sec θ =
r
x
13
–
=
5
csc θ =
r
y
13
–
=
12
cot θ =
x
y
=
5
12
GUIDED PRACTICE
4.
for Examples 1 and 2
Use the unit circle to evaluate the six
trigonometric functions of θ = 180°.
SOLUTION
Draw the unit circle, then draw the angle θ = 180° in
standard position. The terminal side of θ intersects
the unit circle at (–1, 0), so use x = –1 and y = 0 to
evaluate the trigonometric functions.
GUIDED PRACTICE
sin θ =
y
r
0
= 1 =0
tan θ =
y
x
0
= –1
sec θ =
r
–1
= 1 = –1
x
for Examples 1 and 2
cos θ =
x
–1
=
r
1
csc θ =
r
y
–1
= 0
undefined
cot θ =
x
y
–1
= 0
undefined
= –1