Download Intro to Trig

Name:
Intro to Trigonometry
Directions: For the following worksheet, please put only your answers in the spaces shown below. Please remember to
give exact answers, not decimal approximations. Attach scratch paper to show your work.
In order to help you prepare for Test 4, please DO NOT USE A CALCULATOR ON THIS ASSIGNMENT.
1. Section 5.1 Review: A sector has perimeter 15 inches and central angle of 2 radians. Determine its area.
15
Since Perimeter is 2r + s and arclength is s = rθ, we have 15 = 2r + r(2). Thus,
= r.
4
2
1 15
Plug this into the sector area formula to get A =
(2).
2 4
The Six Trigonometric Ratios
Given a right triangle, you will have three side lengths
labeled hypotenuse (HYP), adjacent (ADJ), and opposite (OPP). As shown to the left, notice that the
placement of the angle matters.
In both cases, the angle opposite of the right angle is
the hypotenuse, while the the adjacent side is always
the side touching the angle.
There are six ways to pair up these side lengths, and they are named as follows:
sin(θ) =
OPP
HYP
cos(θ) =
ADJ
HYP
tan(θ) =
OPP
ADJ
cot(θ) =
ADJ
OPP
sec(θ) =
HYP
ADJ
csc(θ) =
HYP
OPP
Verbally, they are sine of theta, cosine of theta, tangent of theta, cotangent of theta, secant of theta, and cosecant of
theta. Also, please get in the habit of using parentheses for angle.
Side notes:
◦ Without the degree symbol, WebAssign (and I) will assume your angle is in radians.
◦ When I mention trig(θ), I mean find all six trig functions for the given angle.
2
. Compute trig(θ).
9
OPP
Using the given information, we can create a triangle using the fact that sin(θ) =
.
HYP
Sample Problem: Suppose sin(θ) =
This gives us two sides, and we can use the Pythagorean Theorem to get the third side.
We can then use this
√ to find the other five trig ratios.
√
ADJ
77
OPP
2
ADJ
77
cos(θ) =
=
tan(θ) =
=√
cot(θ) =
=
HYP
9
ADJ
OPP
2
77
sec(θ) =
HYP
9
=√
ADJ
77
csc(θ) =
HYP
9
=
OPP
2
2. Suppose that tan(θ) = 11. Compute trig(θ).
sin(θ) =
OPP
11
=√
HYP
122
cos(θ) =
ADJ
1
=√
HYP
122
tan(θ) =
OPP
11
=
ADJ
1
HYP
csc(θ) =
=
OPP
√
122
11
√
HYP
122
sec(θ) =
=
ADJ
1
cot(θ) =
ADJ
1
=
OPP
11
3. Use the ratios given in the front of the page, as well as what you know about triangles, to determine if the following
are true or false. Circle your answer.
T
F
(a)
sin(θ) =
1
, as in these two functions are reciprocals.
csc(θ)
T
F
(b)
cos(θ) =
1
, as in these two functions are reciprocals.
sec(θ)
T
F
(c)
tan(θ) =
1
, as in these two functions are reciprocals.
cot(θ)
T
F
(d)
It is possible for sin(θ) to be bigger than 1.
T
F
(e)
It is possible for cos(θ) to be bigger than 1.
T
F
(f)
It is possible for tan(θ) to be bigger than 1.
4. A right triangle has base length of 4 in and base angle 14◦ . Answer the following:
(a) Use trigonometric ratios to determine the height and the hypotenuse of the triangle.
tan(14◦ ) =
a
4
4
, so 4 tan(14◦ ) = a. Also, cos(14◦ ) = , so c =
4
c
cos(14◦ )
(b) Determine the area of the triangle.
Area =
1
1
1
· base · height = · b · a = · 4 · 4 tan(14◦ ).
2
2
2
(c) Determine the perimeter of the triangle.
Perimeter = sum of three sides = 4 + a + c = 4 + 4 tan(14◦ ) +
4
cos(14◦ )
5. A right triangle has hypotenuse length of 8 cm. If one of its interior angles is 23◦ , determine the lengths of its
remaining two sides.
sin(23◦ ) =
a
, so 8 sin(23◦ ) = a.
8
cos(23◦ ) =
b
, so 8 cos(23◦ ) = b
8
6. Use the diagram below to answer the following.
(a) Compute h. Hint: Break up the diagram into a rectangle and a triangle.
h
tan(51◦ ) =
, so 10 tan(51◦ ) = h.
10
(b) Determine the area of the polygon.
Area = Rectangle + Triangle
1
base · height
2
1
= 35 · 10tan(51◦ ) + 10 · 10tan(51◦ )
2
= base · height +