Transcript Alg2 CH 13.3 13.4 - BoxCarChallenge.com

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
EXAMPLE 3
Find reference angles
5π
Find the reference angle θ' for (a) θ =
3
and (b) θ = – 130°.
SOLUTION
a. The terminal side of θ lies in Quadrant IV.
π
5π
So, θ' = 2π –
.
=
3
3
b. Note that θ is coterminal with 230°, whose terminal
side lies in Quadrant III. So, θ' = 230° – 180° + 50°.
EXAMPLE 4
Use reference angles to evaluate functions
Evaluate (a) tan ( – 240°) and (b) csc 17π .
6
SOLUTION
a. The angle – 240° is coterminal
with 120°. The reference angle is
θ' = 180° – 120° = 60°. The tangent
function is negative in Quadrant
II, so you can write:
tan (–240°) = – tan 60° = – √ 3
EXAMPLE 4
Use reference angles to evaluate functions
b. The angle 17π is coterminal
6
5π
with
. The reference
6
5π
π
angle is θ' = π –
=
.
6
6
The cosecant function is
positive in Quadrant II, so you
can write:
5π
csc 17π = csc
=2
6
6
EXAMPLE 5
Calculate horizontal distance traveled
Robotics
The “frogbot” is a robot designed for exploring rough
terrain on other planets. It can jump at a 45° angle and
with an initial speed of 16 feet per second. On Earth,
the horizontal distance d (in feet) traveled by a
projectile launched at an angle θ and with an initial
speed v (in feet per second) is given by:
2
v
d = 32 sin 2θ
How far can the frogbot jump on Earth?
EXAMPLE 5
Calculate horizontal distance traveled
SOLUTION
2
v
Write model for horizontal distance.
d = 32 sin 2θ
2
16
d = 32 sin (2 45°) Substitute 16 for v and 45° for θ.
= 8
Simplify.
The frogbot can jump a horizontal distance of 8 feet
on Earth.
EXAMPLE 1
Evaluate inverse trigonometric functions
Evaluate the expression in both radians and degrees.
a. cos–1 √ 3
2
SOLUTION
a.
When 0 ≤ θ ≤ π or 0°≤ θ ≤ 180°, the angle
whose cosine is √ 3
2
θ =
cos–1
π
√3
=
2
6
–1 √ 3 = 30°
cos
θ =
2
EXAMPLE 1
Evaluate inverse trigonometric functions
Evaluate the expression in both radians and degrees.
b. sin–1 2
SOLUTION
b. There is no angle whose sine is 2. So, sin–1 2 is
undefined.
EXAMPLE 1
Evaluate inverse trigonometric functions
Evaluate the expression in both radians and degrees.
c. tan–1 ( – √ 3 )
SOLUTION
c.
When – π < θ < π , or – 90° < θ < 90°, the
2
2
angle whose tangent is – √ 3 is:
θ=
tan–1
π
( –√ 3 )= –
3
θ = tan–1 ( – √ 3 ) = –60°
EXAMPLE 2
Solve a trigonometric equation
5
Solve the equation sin θ = – 8 where 180° < θ < 270°.
SOLUTION
STEP 1
Use a calculator to determine that in the
interval –90° ≤ θ ≤ 90°, the angle whose
sine is – 5 is sin–1 – 5 – 38.7°. This
8
8
angle is in Quadrant IV, as shown.
EXAMPLE 2
Solve a trigonometric equation
STEP 2
Find the angle in Quadrant III (where
180° < θ < 270°) that has the same sine
value as the angle in Step 1. The angle is:
θ
180° + 38.7° = 218.7°
CHECK : Use a calculator to check the answer.
5
–

sin 218.7° – 0.625=
8
EXAMPLE 3
Standardized Test Practice
SOLUTION
In the right triangle, you are given the lengths of the
side adjacent to θ and the hypotenuse, so use the
inverse cosine function to solve for θ.
cos θ =
6
adj
= 11
hyp
θ = cos
ANSWER The correct answer is C.
–1
6
11
56.9°
EXAMPLE 4
Write and solve a trigonometric equation
Monster Trucks
A monster truck drives off a ramp in order to jump
onto a row of cars. The ramp has a height of 8 feet
and a horizontal length of 20 feet. What is the angle θ
of the ramp?
EXAMPLE 4
Write and solve a trigonometric equation
SOLUTION
STEP 1 Draw: a triangle that represents the ramp.
STEP 2 Write: a trigonometric equation
that involves the ratio of the
ramp’s height and horizontal
length.
8
opp
tan θ =
= 20
adj
EXAMPLE 4
Write and solve a trigonometric equation
STEP 3 Use: a calculator to find the measure of θ.
θ=
tan–1
8
20
21.8°
ANSWER
The angle of the ramp is about 22°.