Algebra 113
First Year of Algebra
Supplementary Unit on Trigonometry
Written by:
Wally Dodge
Pat Wadecki
Kathy Anderson
Becky Lee
David Fickett
Jean Hostetler
(modified)
First Year Algebra 113 Supplemental Trigonometry Section #1
In this section you will learn the following:
A) The definition of the 3 trigonometry ratios: sine (sin), cosine (cos), and tangent (tan)
B) How to use your calculator to find trigonometric ratios
C) How to use trigonometry to solve problems.
Consider the following right triangle.
In mathematics we often use the Greek letter q, pronounced “theta,” to indicate an angle in a triangle. The leg that forms one of the sides of Ðq is called the leg adjacent to Ðq. The other leg is called the leg opposite from Ðq.
Example 1: Complete the chart for each triangle given below.
Table #1
Triangle / Leg opposite q / Leg adjacent q / hypotenuse / Ratio of opposite leg to hypotenuse / Ratio of adjacent leg to hypotenuse / Ratio of opposite leg to adjacent legI
II
III
Solution to example 1 from previous page:
Table #1
Triangle / Leg opposite q / Leg adjacent q / hypotenuse / Ratio of opposite leg to hypotenuse / Ratio of adjacent leg to hypotenuse / Ratio of opposite leg to adjacent legI / 3 / 4 / 5 / 3/5 = 0.6 / 4/5 = 0.8 / 3/4 = 0.75
II / 5 / 12 / 13 / 5/13 ≈ 0.385 / 12/13 ≈ 0.923 / 5/12 ≈ 0.417
III / 15 / 8 / 17* / 15/17 ≈ 0.882 / 8/17 ≈ 0.471 / 15/8 = 1.875
*Note: We used the Pythagorean Theorem for triangle III, in order to find that the hypotenuse was 17.
Using a protractor to measure the angles marked q in each of the triangles labeled I, II, and III on the previous page, we found the information summarized in the table below.
Table #2
Triangle / Measure of qin degrees
I / ≈ 36.9°
II / ≈ 22.6°
III / ≈ 61.9°
Example 2: Set your calculator into “Degree” mode and complete the following table by choosing the appropriate keys on your calculator.
Table #3
Triangle / q in degrees / sin(q) / cos(q) / tan(q)I / 36.9°
II / 22.6°
III / 61.9°
Solution to example 2 from above:
Table #3
Triangle / q in degrees / sin(q) / cos(q) / tan(q)I / 36.9° / ≈ 0.600 / ≈ 0.800 / ≈ 0.751
II / 22.6° / ≈ 0.384 / ≈ 0.923 / ≈ 0.416
III / 61.9° / ≈ 0.882 / ≈ 0.471 / ≈ 1.873
If you compare the columns with the ratios in Table #1 with the sin(q), cos(q), and tan(q) columns in Table #3, you will notice something very interesting. The results are nearly identical. This not a coincidence: these ratios are the definitions of sin(q), cos(q), and tan(q).
Definition: In a right triangle with an acute angle labeled q, the trigonometric ratios sine(q), cosine(q), and tangent(q) are defined as follows:
sine(q), abbreviated sin(q) = , or sin(ÐA) =
cosine(q), abbreviated cos(q) = , or cos(ÐA) =
tangent(q), abbreviated tan(q) = , or tan(ÐA) =
Example 3: Given the figure below, find sin(ÐM ), cos(ÐM), and tan(ÐM)
Solution to Example 3:
By the Pythagorean Theorem, MP = 15
sin(ÐM) = = 0.8
cos(ÐM) = = 0.6
tan(ÐM) = ≈ 1.333
Example 4: For the figure given below of a right triangle with an acute angle of 35°, do the following:
a) Using your knowledge of trigonometric ratios, write two equations, one involving x and the other involving y.
b) Solve the equations that your wrote in part (a).
Solution to Example 4:
a) tan(35°) = and cos(35°) =
Note: if you use the trigonometric ratio for sine you get sin(35°) = which is an equation with both x and y. It is a correct equation, but, since it has both variables in it, this equation isn’t useful to us.
b) Using the trig keys on the calculator (make sure you’re still in “degree” mode!), we obtain:
tan(35°) = cos(35°) =
0.7002075 ≈ .819152 ≈
0.7002075*4 ≈ x .819152*y ≈ 4
0.7002075*4 ≈ x y ≈
2.80 ≈ x y ≈ 4.88
Practice Problems Using Trigonometry (solutions follow on pages 8 and 9)
1. Find the height of the flagpole given the information shown on the diagram.
2. An airplane is 50 miles from an airport and the angle of depression of the airport is 7°. How high is the airplane above the ground?
3. A picture that is 4 feet wide is to be hung from a wire making an angle of 40° with the picture, as shown below. How long is the wire?
4. The Marina is directly across Springstead Lake from the Frontier Inn. To get from one to the other without getting your feet wet, you measure horizontally 4575 feet from the Frontier Inn to point A, and find that the measure of ÐA is 25°. What is the direct distance, across the lake, from the Marina to the Frontier Inn?
SOLUTIONS to Section #1: Practice Problems Using Trigonometry
#1) Let x = the height of the flagpole, in feet. From the diagram, we have:
tan(51°) =
x = 36 * tan(51°)
calculator gives x ≈ 44.46
So the height of the flagpole is approximately 44.46 feet.
#2) The angle of depression is the angle measured from the horizontal to the direction looking downward, as shown in the diagram. (Further explanation on the angle of depression is given on page 9.) Let x = the height of the airplane. From the diagram, we have:
sin(7°) =
x = 50 * sin(7°)
calculator gives x ≈ 6.095
The airplane is approximately 6.095 miles above ground, which is a little over 32,000 feet.
(This is a reasonable cruising altitude for a commercial airliner.)
#3) To use trigonometry, we need a right triangle, so we draw in the height of the triangle formed by the wire. This height divides the horizontal side of the picture into two equal pieces. Let
x = one-half the length of the wire, as shown in the diagram below.
From this diagram (with the height drawn in), we have:
cos(40°) =
x * cos(40°) = 2
x = ≈ 2.61
Since x represents half the length of the wire, the wire is approximately 5.22 feet long.
More SOLUTIONS to Section #1: Practice Problems Using Trigonometry
#4) Let x = the distance, in feet, across the lake from the Marina to the Inn, as shown below.
From this diagram, we have:
tan(25°) =
x = 4575 * tan(25°)
x ≈ 2133.36
The distance across the lake from the Marina to the Frontier Inn is approximately 2133 feet.
NOTE: For some applications of trigonometry, you need to know the meanings of angle of elevation and angle of depression.
If an observer at point P looks upward toward an object at A, the angle the line of sight makes with the horizontal is called the angle of elevation.
If an observer at point P looks downward toward an object at B, the angle the line of sight makes with the horizontal is called the angle of depression.
(from: Rhoad, Milauskus, & Whipple: Geometry for Enjoyment and Challenge, New Edition. McDougal, Little, & Company: 1991)
In class Example:
1. Tom is looking up at his friend, Jen who is in a tree. He estimates his angle of elevation to be 40⁰ and is standing 20 feet away from the base of the tree. How high off the ground is Jen, to the nearest foot?
2. Jack is looking out the window at Jill. He estimates his angle of depression to be 30⁰. He know he’s about 45 feet above ground level. How far is Jill from the building?
Exercises for Trigonometry: Section I
1) a) Label the sides of the triangle opposite, hypotenuse,
and adjacent using ÐA as the given angle.
b) Label the sides of the triangle opposite, hypotenuse,
and adjacent using ÐB as the given angle.
2) For the given triangles, with angle q, do the following:
i) Give the lengths of the side adjacent to q, the side opposite q, and the hypotenuse.
ii) Give the values of sin(q), cos(q), and tan(q).
a) b)
c) d)
3) Find sin(ÐA), cos(ÐA), and tan(ÐA) in terms of a, b, and c.
4) Using your knowledge of trigonometry, find the length of the missing side marked x on each figure.
a) b)
c) d)
5) Find the value of x in each case.
a) b) c)
6) Given a right triangle ABC with the right angle at C.
a) Draw a possible picture of this right triangle.
b) If sin(ÐA) = 3/5, then cos(ÐA) =
c) If tan(ÐA) = 2/3, then sin(ÐA) = (info from previous parts no longer true)
d) If cos(ÐA) = 4/7, then tan(ÐA) = (info from previous parts no longer true)
7) Round your answers to this problem to the nearest hundredth. (Use the triangle shown!)
Find: a) sin(30°) d) cos(30°)
b) cos(60°) e) tan(30°)
c) sin(60°) f) tan(60°)
8) Round your answers to the nearest hundredth.
Find: a) sin(45°) b) cos(45°) c) tan(45°)
d) Why are sin(45°) and cos(45°) the same?
9) You are flying a kite on a string that is 150 feet in length. The angle that the kite string makes with the ground is 75°. Find the height of the kite above the ground.
10) You are parasailing behind a boat. You are 125 feet above the water, and the rope that you are holding (attached to the boat) makes an angle of 50° with the water. How long is the rope that you are holding?
11) To find the distance across Lake Springstead without getting his feet wet, Sam the surveyor measures the distance (3600 feet) and the angle (22°) shown in the diagram below. If ÐB is a right angle, how far is it across Lake Springstead?
12) You are 60 feet due west from the front of the Eurostar train in Waterloo station, which is facing North/South. You calculate an angle of 60° between the line from where you are standing to the front of the train and the line from where you are standing to the rear of the train. How long is the train?
13) You are at an elevation of 18,500 feet and you spot the peak
of Mt. McKinley about one mile away along your line of sight.
You can’t remember how high Mt. McKinley is, so you take out
some of your surveying equipment to find the angle of elev.
You find that the angle of elevation of the peak of Mt.McKinley,
from where you are standing, is approximately 20.2°.
How high is this mountain peak?
14) A guard standing on the lookout on top of a castle 250 feet high has spotted an army of knights approaching at an angle of depression of 2°. How far is this army from the castle?
15) Heimlich maneuvered his mountain bike straight up the side of a hill with a 16° grade. If the hill had an elevation (= height) of 1600 feet, how long was the trail up the hill?
16) Mario parked his car at a 45° angle with the curb,
as shown in the diagram. If the car is 14 feet long
and 6 feet wide, how far is the right rear corner of
the car from curb? How far is the left front corner
from the curb?
17) Mrs. Bluebird is on a mad hunt for a worm to give to her babies for breakfast. She is flying at an altitude of 34 feet when she spots a worm at an angle of depression of 19°. How far must she travel to catch breakfast for her babies?
18) Rectangle ABCD is inscribed in circle O. (That is, it just fits inside the circle, and it’s corners are labeled A, B, C, D as you walk around the circle.) If the radius of circle O is 19 and the measure of ÐBDC is 30°, find the perimeter of triangle ABD.
19) A satellite hovers 300 km above the planet. The angle from the satellite to the horizon is 80°,
as shown in the diagram. What is the radius of the planet?
First Year Algebra 113 Supplemental Trigonometry Section #2
In this section you will learn the following:
A) How to find an angle when given a trigonometric ratio
B) How to use your calculator to find angles
C) How to use trigonometry to do application problems.
In the last section you learned how to find lengths of sides in a right triangle when given one side and an acute angle in the triangle. In this section you will learn how to find an acute angle in a right triangle when given two sides of that triangle.
Example 1: Use your graphing calculator’s “table” tool to find an approximate value for the angle shown in each triangle below.
Solution For Triangle I: tan(q) = ≈ 0.5714; in “degree” mode, let y1 = tan(x) and set up a table to start at 0° and move in steps of 1°, as shown below. To the nearest degree, q ≈ 30°.