Download 9.1 Use Trigonometry with Right Triangles

9.1 Use Trigonometry with
Right Triangles
• Use the Pythagorean Theorem to find
missing lengths in right triangles.
• Find trigonometric ratios using right
triangles.
• Use trigonometric ratios to find angle
measures in right triangles.
• Use special right triangles to find lengths
in right triangles.
Pythagorean Theorem
In a right triangle, the square of the length
of the hypotenuse is equal to the sum of
the squares of the lengths of the legs.
c
a2 + b2 = c2
a
b
In right triangle ABC, a and b are the lengths of the
legs and c is the length of the hypotenuse. Find
the missing length. Give exact values.
1. a = 6, b = 8
ANSWER c = 10
2. c = 10, b = 7
ANSWER a =
51
Definition
Trigonometry—the study of the special
relationships between the angle
measures and the side lengths of right
triangles.
A trigonometric ratio is the ratio the
lengths of two sides of a right triangle.
Right triangle parts.
B
hypotenuse
leg
A
C
leg
Name the hypotenuse. AB
Name the legs.
BC , AC
The Greek letter theta 𝜽
Opposite or Adjacent?
west
• The opposite of east is ______
• The door is opposite the windows.
• In a ROY G BIV, red is adjacent to orange
_______
• The door is adjacent to the cabinet.
Triangle Parts
B
hypotenuse
c
A
Opposite side
Adjacent side
a
C
b
Opposite
Adjacent
side
side
Looking from angle B:
Which side is the opposite? b
Which side is the adjacent? a
Looking from angle A:
Which side is the hypotenuse? c
Which side is the opposite?
a
Which side is the adjacent?
b
Trigonometric ratios
B
BC a
length of leg opposite A

sine of A 

AB c
length of hypotenuse
cosecant of ∠𝐴 =
hypotenuse
𝐴𝐵
=
length of leg opposite 𝐵𝐶
=
c
a
𝑐
𝑎
A
length of leg adjacent A AC b

cosine of A 

AB c
length of hypotenuse
hypotenuse
𝐴𝐵 𝑐
secant of ∠𝐴 =
=
=
length of leg adjacent ∠𝐴 𝐴𝐶 𝑏
b
BC a
length of leg opposite A

tangent of A 

AC b
length of leg adjacent to A
length of the hypotenuse 𝐴𝐵 𝑐
cotangent of ∠𝐴 =
=
=
length of leg opposite ∠𝐴 𝐵𝐶 𝑎
C
Trigonometric ratios and definition
abbreviations
opposite
a
c
sin A 

hypotenuse c
adjacent
b
cos A 

hypotenuse c
opposite a
tan A 

adjacent b
SOHCAHTOA
A
b
B
a
C
Trigonometric ratios and definition
abbreviations
c
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑐𝑠𝑐 ∠𝐴 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
s𝑒𝑐 ∠𝐴 =
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝑐𝑜𝑡∠𝐴 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
A
1
𝑐𝑠𝑐 =
𝑠𝑖𝑛
𝑠𝑖𝑛−1
1
𝑠𝑒𝑐 =
𝑐𝑜𝑠
𝑐𝑜𝑠 −1
1
𝑐𝑜𝑡 =
𝑡𝑎𝑛
𝑡𝑎𝑛−1
B
a
b
C
Evaluate the six trigonometric functions of the
angle θ.
1.
SOLUTION
From the Pythagorean theorem, the length of the
hypotenuse is √ 32 + 42 = √ 25 = 5.
sin θ =
opp
=
hyp
cos θ =
adj
4
= 5
hyp
tan θ =
opp
=
adj
3
5
3
4
csc θ =
hyp
5
=
3
opp
sec θ =
hyp
5
= 4
adj
cot θ =
adj
4
=
opp
3
Evaluate the six trigonometric functions of the angle θ.
3.
SOLUTION
From the Pythagorean theorem, the length of the
adjacent is 5 2 2 − 52 = 25 = 5.
sin θ =
opp
5
=
hyp
5√2
cos θ =
adj
5
=
hyp
5√2
tan θ =
opp
=
adj
5
=1
5
csc θ =
hyp
5√2
=
5
opp
sec θ =
hyp
5√2
=
5
adj
cot θ =
adj
5
=1
=
opp
5
45°- 45°- 90° Right Triangle
In a 45°- 45°- 90° triangle, the hypotenuse is √2
times as long as either leg. The ratios of the
side lengths can be written l-l-l√2.
l 2
l
l
30°- 60° - 90° Right Triangle
In a 30°- 60° - 90° triangle, the hypotenuse is twice
as long as the shorter leg (the leg opposite the
30° angle, and the longer leg (opposite the 60°
angle) is √3 tunes as long as the shorter leg. The
ratios of the side lengths can be written l - l√3 –
2l.
60°
2l
l
30°
l 3
4. In a right triangle, θ is an acute angle and cos
θ = 7 . What is sin θ?
10
SOLUTION
STEP 1
Draw: a right triangle with acute
angle θ such that the leg opposite
θ has length 10 and the
hypotenuse has length 7. By the
Pythagorean theorem, the length x
of the other leg is x = √ 102 – 72 = √ 51.
STEP 2
Find: the value of sin θ.
sin θ =
opp
√ 51
= 10
hyp
ANSWER
sin θ =
√ 51
10
Find the value of x for the
right triangle shown.
SOLUTION
Write an equation using a trigonometric function that
involves the ratio of x and 8. Solve the equation for x.
adj
cos 30º =
Write trigonometric equation.
hyp
√3
x
=
8
2
4√ 3 = x
Substitute.
Multiply each side by 8.
ANSWER
The length of the side is x = 4 √ 3
6.93.
Make sure your
calculator is set to
degrees.
ABC.
Solve
SOLUTION
A and B are complementary angles,
so B = 90º – 28º = 68º.
tan 28° =
opp
adj
a
tan 28º =
15
15(tan 28º) = a
7.98
a
ANSWER
cos 28º =
adj
hyp
15
cos 28º =
c
c∙ 𝑐𝑜𝑠28° = 15
15
𝑐=
𝑐𝑜𝑠28°
c
17.0
So, B = 62º, a
Write trigonometric
equation.
Substitute.
Solve for the
variable.
Use a calculator.
7.98, and c
17.0.
Solve ABC using the diagram at the
right and the given measurements.
5. B = 45°, c = 5
SOLUTION A and B are complementary angles,
so A = 90º – 45º = 45º.
adj
cos 45° =
sin 45º =
hyp
a
cos 45º =
5
5(cos 45º) = a
3.54
a
ANSWER
opp
hyp
b
5
5(sin 45º) = b
sin 45º =
3.54
So, A = 45º, b
b
Write trigonometric
equation.
Substitute.
Solve for the variable.
Use a calculator.
3.54, and a
3.54.
6. A = 32°, b = 10
SOLUTION
A and B are complementary angles,
so B = 90º – 32º = 58º.
opp
hyp
tan 32° =
sec 32º =
adj
adj
a
tan 32º =
10
10(tan 32º) = a
6.25
a
ANSWER
sec 32º = c
10
Write trigonometric
equation.
Substitute.
Solve for the variable.
11.8
c
So, B = 58º, a
Use a calculator.
6.25, and c
11.8.
7.
A = 71°, c = 20
SOLUTION
A and B are complementary angles,
so B = 90º – 71º = 19º.
adj
opp
cos 71° =
sin 71º =
hyp
hyp
b
cos 71º =
20
20(cos 71º) = b
6.51
ANSWER
b
a
sin 71º =
20
20(sin 71º) = a
18.9
a
So, B = 19º, b
Write trigonometric
equation.
Substitute.
Solve for the variable.
Use a calculator.
6.51, and a
18.9.
8.
B = 60°, a = 7
SOLUTION
A and B are complementary angles,
so A = 90º – 60º = 30º.
hyp
opp Write trigonometric
sec 60° =
tan 60º =
adj
adj
equation.
c
7
sec 60º =
)
(
1
7
=c
cos 60º
14 = c
ANSWER
b
tan 60º =
7
7(tan 60º) = b
12.1
b
Substitute.
Solve for the variable.
Use a calculator.
So, A = 30º, c = 14, and b
12.1.
Parasailing
A parasailer is attached to a boat with
a rope 300 feet long. The angle of
elevation from the boat to the
parasailer is 48º. Estimate the
parasailer’s height above the boat.
SOLUTION
STEP 1 Draw: a diagram that represents the situation.
STEP 2 Write: and
h solve an equation to find the height h.
sin 48º =
Write trigonometric equation.
300
300(sin 48º) = h
Multiply each side by 300.
223 ≈ x
Use a calculator.
ANSWER
The height of the parasailer above the
boat is about 223 feet.
• Use the Pythagorean Theorem to find missing
lengths in right triangles.
𝑎2 + 𝑏2 = 𝑐 2
• Find trigonometric ratios using right triangles.
Since, cosine, tangent, cotangent, secant,
cosecant
• Use trigonometric ratios to find angle
measures in right triangles.
• Use special right triangles to find lengths
in right triangles.
In a 45°-45°-90° right triangle, use 𝑙, 𝑙, 𝑙 2
In a 30°-60°-90° right triangle, use 𝑙, 𝑙 2, 2𝑙
9.1 Assignment
Page 560, 4-14 even, 17, 19,
22-26 even,
skip 8