9.1 use trigonometry with right triangles find trigonometric ratios using right triangles. use...
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9.1 Use Trigonometry with Right Triangles
• Find trigonometric ratios using right triangles.
• Use trigonometric ratios to find angle measures in right triangles.
• Use the Pythagorean Theorem to find missing lengths in right triangles.
• Use special right triangles to find lengths in right triangles.
Pythagorean TheoremIn 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.
a2 + b2 = c2
b
ac
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
2. c = 10, b = 7
ANSWER c = 10
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.
AB
leg
leghypotenuse
A
B
C
Name the hypotenuse.
Name the legs. ACBC ,
The Greek letter theta
Opposite or Adjacent?
• The opposite of east is ______• The door is opposite the windows.
• In a ROY G BIV, red is adjacent to _______• The door is adjacent to the cabinet.
west
orange
Triangle Parts
A
B
C
a
b
c
Opposite sideAdjacent side
hypotenuseOpposite side
Adjacent side
Looking from angle A:
Which side is the hypotenuse?
Which side is the opposite?
Which side is the adjacent?
c
a
b
Looking from angle B:
Which side is the opposite?
Which side is the adjacent?
b
a
Trigonometric ratios
hypotenuseoflength
Aoppositelegoflengthsine Aof
c
a
AB
BC
A
B
C
a
b
c
hypotenuseoflength
Aadjacentlegoflengthcosine Aof
c
b
AB
AC
Atoadjacentlegoflength
Aoppositelegoflengthtangent Aof
b
a
AC
BC
c otangent of ∠𝐴=lengthof the hypotenuselength of leg opposite∠𝐴
=𝐴𝐵𝐵𝐶
=𝑐𝑎
secant of∠ 𝐴=hypotenuse
length of leg adjacent∠ 𝐴=𝐴𝐵𝐴𝐶
=𝑐𝑏
=
Trigonometric ratios and definition abbreviations
c
aA
hypotenuse
oppositesin
c
bA
hypotenuse
adjacentcos
A
B
C
a
b
c
b
aA
adjacent
oppositetan
SOHCAHTOA
Trigonometric ratios and definition abbreviations
A
B
C
a
b
c
𝑐𝑠𝑐∠ 𝐴=h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
s
𝑐𝑜𝑡∠ 𝐴=𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑐𝑠𝑐=1𝑠𝑖𝑛
𝑠𝑒𝑐=1
𝑐𝑜𝑠
𝑐𝑜𝑡=1𝑡𝑎𝑛
𝑠𝑖𝑛−1
𝑐𝑜𝑠− 1
𝑡𝑎𝑛− 1
Evaluate the six trigonometric functions of the angle θ. 1.SOLUTION
5.= 25= √From the Pythagorean theorem, the length of thehypotenuse is 32 + 42√
sin θ =opphyp =
35
csc θ =hypopp =
53
tan θ =oppadj =
34
cot θ =adjopp =
43
cos θ =adjhyp =
45 sec θ =
hypadj =
54
Evaluate the six trigonometric functions of the angle θ.
SOLUTION
sin θ =opphyp
csc θ =hypopp
3.
From the Pythagorean theorem, the length of theadjacent is
=5
5 2√ 5=5 2√
√ (5 √2 )2−52=√25=5.
tan θ =oppadj =
55
cot θ =adjopp =
55
cos θ =adjhyp
sec θ =hypadj
5= 5 2√ 5=
5 2√
= 1 = 1
45°- 45°- 90° Right TriangleIn 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
l
2l
30°- 60° - 90° Right TriangleIn 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°
30°
l
2l
3l
4. In a right triangle, θ is an acute angle and cos θ = . What is sin θ?7
10
SOLUTION
STEP 1
51.= √
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 102 – 72√x =
STEP 2 Find: the value of sin θ.
sin θ =opphyp =
51√10
ANSWER
sin θ =51√
10
SOLUTION
Write an equation using a trigonometric function that involves the ratio of x and 8. Solve the equation for x.
Find the value of x for the right triangle shown.
cos 30º =adjhyp Write trigonometric equation.
32
√ = x8 Substitute.
34 √ = x Multiply each side by 8.
The length of the side is x = 34 √ 6.93.
ANSWER
SOLUTION
Write trigonometric equation.
Substitute.
Solve ABC.
A and B are complementary angles,so B = 90º – 28º
tan 28° =oppadj cos 28º =
adjhyp
tan 28º =a
15 cos 28º =15c
= 68º.
Solve for the variable.
Use a calculator.
15(tan 28º) = a
7.98 a 17.0 c
So, B = 62º, a 7.98, and c 17.0.ANSWER
c
𝑐=15
𝑐𝑜𝑠28 °
Make sure your calculator is set to degrees.
Solve ABC using the diagram at the right and the given measurements.
5. B = 45°, c = 5SOLUTION
Substitute.
A and B are complementary angles,
so A = 90º – 45º
cos 45° =adjhyp sin 45º =
opphyp
cos 45º =a
5 sin 45º = 5b
Write trigonometric equation.
= 45º.
Solve for the variable.
Use a calculator.
5(cos 45º) = a 5(sin 45º) = b
3.54 a 3.54 b
So, A = 45º, b 3.54, and a 3.54.ANSWER
SOLUTION
Substitute.
A and B are complementary angles,so B = 90º – 32º
tan 32° =oppadj sec 32º =
hypadj
tan 32º =a
10 sec 32º = 10c
6. A = 32°, b = 10
Write trigonometric equation.
= 58º.
Solve for the variable.
Use a calculator.
10(tan 32º) = a
6.25 a 11.8 c
So, B = 58º, a 6.25, and c 11.8.ANSWER
SOLUTION
Substitute.
A and B are complementary angles,so B = 90º – 71º
cos 71° =adjhyp sin 71º =
opphyp
cos 71º =b
20 sin 71º =a
20
7. A = 71°, c = 20
Write trigonometric equation.
= 19º.
Solve for the variable.
Use a calculator. 20(cos 71º) = b
6.51 b 18.9 a
So, B = 19º, b 6.51, and a 18.9.ANSWER
20(sin 71º) = a
SOLUTION
Substitute.
A and B are complementary angles,so A = 90º – 60º
sec 60° =hypadj tan 60º =
oppadj
sec 60º = 7c tan 60º =
b7
Write trigonometric equation.
8. B = 60°, a = 7
= 30º.
Solve for the variable.
Use a calculator.
7(tan 60º) = b 7 1(cos 60º
)= c
14 = c 12.1 b
So, A = 30º, c = 14, and b 12.1.ANSWER
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.
Parasailing
SOLUTION
sin 48º =h
300 Write trigonometric equation.
300(sin 48º) = h Multiply each side by 300.
STEP 1 Draw: a diagram that represents the situation.
STEP 2 Write: and solve an equation to find the height h.
223 ≈ x Use a calculator.
The height of the parasailer above the boat is about 223 feet.
ANSWER
• Find trigonometric ratios using right triangles.
• Use trigonometric ratios to find angle measures in right triangles.
• Use the Pythagorean Theorem to find missing lengths in right triangles.
• Use special right triangles to find lengths in right triangles.
𝑎2+𝑏2=𝑐2
Since, cosine, tangent, cotangent, secant, cosecant
In a 45°-45°-90° right triangle, use
In a 30°-60°-90° right triangle, use
9.1 Assignment
Page 560, 4-14 even, 17, 19, 22-26 even,
skip 8