Transcript Alg2 CH 13.1 13.2 - BoxCarChallenge.com

EXAMPLE 1
Evaluate trigonometric functions
Evaluate the six trigonometric functions of the angle θ.
SOLUTION
From the Pythagorean theorem, the length of the
hypotenuse is √ 52 + 122 = √ 169 = 13.
sin θ =
opp
12
=
hyp
13
csc θ =
hyp
13
= 12
opp
EXAMPLE 1
Evaluate trigonometric functions
cos θ =
adj
=
hyp
5
13
tan θ =
opp
=
adj
12
5
sec θ =
13
hyp
= 5
adj
cot θ =
5
adj
= 12
opp
EXAMPLE 2
Standardized Test Practice
SOLUTION
STEP 1
Draw: a right triangle with acute
angle θ such that the leg opposite
θ has length 4 and the hypotenuse
has length 7. By the Pythagorean
theorem, the length x of the other
leg is x = √ 72 – 42 = √ 33.
EXAMPLE 2
Standardized Test Practice
STEP 2
Find the value of tan θ.
tan θ =
4
opp
=
adj
√ 33
4 √ 33
=
33
ANSWER
The correct answer is B.
EXAMPLE 3
Find an unknown side length of a right triangle
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
Write trigonometric equation.
cos 30º =
hyp
√3
2
=
x
8
Substitute.
EXAMPLE 3
4√3
= x
Find an unknown side length of a right triangle
Multiply each side by 8.
ANSWER
The length of the side is x = 4 √ 3
6.93.
EXAMPLE 4
Solve
Use a calculator to solve a right triangle
ABC.
SOLUTION
A and B are complementary angles,
so B = 90º – 28º = 68º.
opp
tan 28° =
adj
a
tan 28º =
15
sec 28º =
hyp
adj
c
sec 28º =
15
Write trigonometric
equation.
Substitute.
EXAMPLE 4
15(tan 28º) = a
7.98
a
Use a calculator to solve a right triangle
15
(
1
cos 28º
17.0
)=c
c
ANSWER
So, B = 62º, a
7.98, and c
17.0.
Solve for the variable.
Use a calculator.
EXAMPLE 5
Use indirect measurement
Grand Canyon
While standing at Yavapai Point near the
Grand Canyon, you measure an angle of
90º between Powell Point and Widforss
Point, as shown. You then walk to Powell
Point and measure an angle of 76º
between Yavapai Point and Widforss
Point. The distance between Yavapai
Point and Powell Point is about 2 miles.
How wide is the Grand Canyon between
Yavapai Point and Widforss Point?
EXAMPLE 5
Use indirect measurement
SOLUTION
tan 76º =
x
2
2(tan 76º) = x
8.0 ≈ x
ANSWER
The width is about 8.0 miles.
Write trigonometric equation.
Multiply each side by 2.
Use a calculator.
EXAMPLE 6
Use an angle of elevation
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.
EXAMPLE 6
Use an angle of elevation
SOLUTION
STEP 1
Draw: a diagram that represents the situation.
STEP 2
Write: and solve an equation to find the height h.
h
Write trigonometric equation.
sin 48º =
300
300(sin 48º) = h
Multiply each side by 300.
Use a calculator.
223 ≈ x
ANSWER
The height of the parasailer above the boat is about 223 feet.
EXAMPLE 1
Draw angles in standard position
Draw an angle with the given measure in standard
position.
a. 240º
SOLUTION
a.
Because 240º is 60º more
than 180º, the terminal side
is 60º counterclockwise past
the negative x-axis.
EXAMPLE 1
Draw angles in standard position
Draw an angle with the given measure in standard
position.
b.
500º
SOLUTION
b.
Because 500º is 140º more
than 360º, the terminal side
makes one whole revolution
counterclockwise plus 140º
more.
EXAMPLE 1
Draw angles in standard position
Draw an angle with the given measure in standard
position.
c.
–50º
SOLUTION
c.
Because –50º is negative, the
terminal side is 50º clockwise
from the positive x-axis.
EXAMPLE 2
Find coterminal angles
Find one positive angle and one negative angle that are
coterminal with (a) –45º and (b) 395º.
SOLUTION
There are many such angles, depending on what
multiple of 360º is added or subtracted.
a.
–45º + 360º = 315º
–45º – 360º = – 405º
EXAMPLE 2
Find coterminal angles
b. 395º – 360º = 35º
395º – 2(360º) = –325º
EXAMPLE 3
Convert between degrees and radians
Convert (a) 125º to radians and (b) –
degrees.
π radians
a. 125º = 125º
180º
)
(
25π
36
=
π
b. –
12
=
radians
π
–
radians
12
(
= –15º
π radians to
12
)(
180º
π radians
)
EXAMPLE 4
Solve a multi-step problem
Softball
A softball field forms a sector with the dimensions
shown. Find the length of the outfield fence and the
area of the field.
EXAMPLE 4
Solve a multi-step problem
SOLUTION
STEP 1
Convert the measure of the central angle to radians.
90º = 90º
(
π radians
180º
)=
π
2
radians
EXAMPLE 4
Solve a multi-step problem
STEP 2
Find the arc length and the area of the sector.
Arc length: s = r θ = 180 ( π ) = 90π ≈ 283 feet
2
Area: A =
25,400 ft2
1 2
rθ=
2
1
2
(180)
2
π
( 2 )= 8100π ≈
ANSWER
The length of the outfield fence is about 283 feet.
The area of the field is about 25,400 square feet.