Download §6-6 RIGHT TRIANGLE TRIGONOMETRY

§6-6
RIGHT TRIANGLE TRIGONOMETRY
In this section we consider the trigonometry of right triangles. The trigonometric
functions of concern are the sine, cosine and tangent functions.
Definition
For a right triangle, the sine of an angle θ is the ratio of the length of the leg opposite θ
opposite leg
to the length of the hypotenuse: sin θ =
.
hypotenuse
Definition
For a right triangle, the cosine of an angle θ is the ratio of the length of the leg adjacent
adjacent leg
θ to the length of the hypotenuse: cos θ =
.
hypotenuse
Definition
For a right triangle, the tangent of an angle θ is the ratio of the length of the leg
opposite leg
opposite θ to the length of the leg adjacent θ : tan θ =
.
adjacent leg
The mnemonic SOH CAH TOA is often used to recall the relationship of the sides of
the right triangle to the trigonometric functions.
Example 1
Find the sine, cosine and tangent of θ and α in the right triangle shown below.
α
5
3
θ
4
Solution
For θ the opposite leg has length 3 and the adjacent leg has length 4. Therefore...
sin θ =
opposite leg 3
=
hypotenuse 5
cos θ =
adjacent leg 4
=
hypotenuse 5
tan θ =
opposite leg 3
=
adjacent leg 4
For α the opposite leg has length 4 and the adjacent leg has length 3. Therefore...
sin α =
opposite leg 4
=
hypotenuse 5
cos α =
adjacent leg 3
=
hypotenuse 5
tan α =
opposite leg 4
=
adjacent leg 3
Copyright©2007 by Lawrence Perez and Patrick Quigley
Example 2
If cos θ = x for the triangle below, then find the length of AC.
A
θ
3
B
adjacent leg
hypotenuse
3
x =
AC
ACx = 3
3
AC =
x
Solution
cos θ =
Identity
2
2
For any angle θ, sin θ + cos θ = 1 .
Example 3
If sin θ =
Solution
sin θ + cos θ = 1
2
2
cos θ = 1 − sin θ
2
2
⎛ 1⎞
cos θ = 1 −
⎝ 2⎠
3
2
cos θ =
4
3
cos θ = ±
2
Identity
For any angle θ, tan θ =
Example 4
If cos θ =
Solution
C
2
1
then find the cosine of θ.
2
2
sin θ
.
cosθ
5
12
and tan θ =
then find the sine of θ.
13
5
sin θ
cos θ
tan θ ⋅ cos θ = sin θ
12 5
⋅
= sin θ
5 13
12
= sin θ
13
tan θ =
Copyright©2007 by Lawrence Perez and Patrick Quigley
§6-6
PROBLEM SET
Find each quantity for the right triangle below.
θ
13
5
α
12
1.
sin α
2.
cos α
3.
tan α
4.
sin θ
5.
cos θ
6.
tan θ
Find the lengths of the missing sides of the right triangle below.
y
cos θ =
x
2
3
tan θ =
θ
5
2
10
7.
Find x.
8.
Find y.
Use the given quantities to calculate the missing quantity.
If sin θ =
1
3
and cos θ =
then find tan θ.
2
2
10. If sin θ =
2
and tan θ = 1 then find cos θ.
2
11. If cos θ =
1
and tan θ = 2 2 then find sin θ .
3
12. If cos θ =
5
and sin θ > 0 then find sin θ .
3
9.
Copyright©2007 by Lawrence Perez and Patrick Quigley
§6-6
PROBLEM SOLUTIONS
1.
5
13
2.
12
13
3.
5
12
4.
12
13
5.
5
13
6.
12
5
7.
15
8.
5 5
9.
3
3
11.
2 2
3
12.
2
3
10.
2
2
Copyright©2007 by Lawrence Perez and Patrick Quigley