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Trigonometry Review
Professor D. Olles
July 7, 2014
Since trigonometry means ”the study of triangles”, let’s begin with the two
special triangles that we use when evaluating trigonometric functions.
1. The 30◦ : 60◦ : 90◦ trigangle:
Consider an equilateral triangle of side lengths 2. Since the equilateral triangle has sides of the same length, the three angles must all be the same
measure. The sum of the measures of the angles in a triangle is 180◦ , so
if we divide that into 3 equally measured angles, we have 60◦ for each. As
shown below.
Fill in the side length on the triangle below. Then, use the bisecting line segment AD to divide the triangle into two equivallent
right triangles. Sketch one of them and fill in all missing sides
and angles, including both degree and radian measure. Notice
that the result is one of the two special triangles we are familiar
with.
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2. The 45◦ : 45◦ : 90◦ triangle:
Any isoceles triangle has two sides that are congruent, with corresponding
congruent angles. Consider a right isoceles triangle. One angle would
then have to be 90◦ leaving the other two to be divided equally over the
remaining 90◦ alloted, giving us 45◦ each. The congruent sides can be of
any length, but √
if consider the simple case of length 1, we are left with a
hypothenuse of 2.
√
Show how the hypothenuse becomes 2 and sketch this triangle.
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Now, let’s exmaine how the unit circle is generated from these triangles. If
we embed the right triangle inside of a circle, in the first quadrant, so that
the hypothenuse of the triangle represents the radius of the circle, we have the
circles below.Fill in the coordinate point where the circle and triangle
intersect.
Since we would like to exmaine the unit circle first, we need the radius to be
equal to 1. So, the above three representations can be redrawn as below. Fill
in the coordinate point where the circle and triangle intersect.
If we consider the resultant first quadrant for the unit circle, and generate the
remaining using symmetry, we are able to achieve:
Another way we may examine the ”slices” of the unit circle and how the angles
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are generated in quadrants II, III, and IV is to consider the upper half of the
circle’s measure of π and divide it accordingly. First, let’s consider the 45◦ or
π
4 slicing. We need to divide π units into 4 equal parts, so we have:
Complete the 3rd and 4th quadrants of this circle, including the coordinates for each angle. Then, repeat the entire process for the π6
division.
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Now, let’s examine the 6 trigononmetric functions and their evaluations at the
special angles.
1. The Sine Function
(a) In Right Triangle Trigonometry:
sin θ =
OPP
HYP
(b) In the Unit Circle:
Below we have a circle with radius r, inside of which we have inscribed
a right triangle. Based on our right triangle trigonometry we have:
sin θ =
y
r
In the Unit Circle, r = 1, so that gives us:
sin θ = y
(c) A table of values for Sine:
Use the Unit Circle and the information above to fill in
the following table of values for the first quadrant. Note
that the remaining quadrant values can be determined using
symmetry.
θ
0
π
6
sin θ
(d) The Sine Curve as a Function:
5
π
4
π
3
π
2
2. The Cosine Function
(a) In Right Triangle Trigonometry:
cos θ =
ADJ
HYP
(b) In the Unit Circle:
Below we have a circle with radius r, inside of which we have inscribed
a right triangle. Based on our right triangle trigonometry we have:
x
cos θ =
r
In the Unit Circle, r = 1, so that gives us:
cos θ = x
(c) A table of values for Cosine:
Use the Unit Circle and the information above to fill in
the following table of values for the first quadrant. Note
that the remaining quadrant values can be determined using
symmetry.
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θ
0
π
6
π
4
π
3
π
2
cos θ
(d) The Cosine Curve as a Function:
3. The Tangent Function
(a) In Right Triangle Trigonometry:
tan θ =
OPP
ADJ
(b) In the Unit Circle:
tan θ =
y
x
(c) A table of values for Tangent:
Use the Unit Circle and the information above to fill in
the following table of values for the first quadrant. Note
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that the remaining quadrant values can be determined using
symmetry.
θ
0
π
6
π
4
π
3
π
2
tan θ
(d) The Tangent Curve as a Function:
4. The Cosecant Function
(a) In Right Triangle Trigonometry:
csc θ =
HYP
OPP
(b) In the Unit Circle:
Below we have a circle with radius r, inside of which we have inscribed
a right triangle. Based on our right triangle trigonometry we have:
r
csc θ =
y
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In the Unit Circle, r = 1, so that gives us:
csc θ =
1
y
(c) A table of values for Cosecant:
Use the Unit Circle and the information above to fill in
the following table of values for the first quadrant. Note
that the remaining quadrant values can be determined using
symmetry.
θ
0
π
6
π
4
π
3
π
2
csc θ
(d) The Cosecant Curve as a Function:
5. The Secant Function
(a) In Right Triangle Trigonometry:
sec θ =
HYP
ADJ
(b) In the Unit Circle:
Below we have a circle with radius r, inside of which we have inscribed
a right triangle. Based on our right triangle trigonometry we have:
sec θ =
9
r
x
In the Unit Circle, r = 1, so that gives us:
sec θ =
1
x
(c) A table of values for Secant:
Use the Unit Circle and the information above to fill in
the following table of values for the first quadrant. Note
that the remaining quadrant values can be determined using
symmetry.
θ
0
π
6
sec θ
(d) The Secant Curve as a Function:
6. The Cotangent Function
10
π
4
π
3
π
2
(a) In Right Triangle Trigonometry:
cot θ =
ADJ
OPP
(b) In the Unit Circle:
cot θ =
x
y
(c) A table of values for Cotangent:
Use the Unit Circle and the information above to fill in
the following table of values for the first quadrant. Note
that the remaining quadrant values can be determined using
symmetry.
θ
0
π
6
csc θ
(d) The Cotangent Curve as a Function:
11
π
4
π
3
π
2
Summary of Right Triangle Trigonometry:
Fill in the following ratios to complete the formulas for evaluating the
trigonometric functions using Right Triangle Trigonometry.
sin θ =
csc θ =
cos θ =
sec θ =
tan θ =
cot θ =
Summary Table of Values for Quadrant I:
Fill in the table using the following procedure:
1. List the values 0 to 4 from left to right in the sin θ row, and do
the opposite in the cos θ row.
θ
0
π
6
π
4
π
3
π
2
sin θ
cos θ
2. Take the square root of every value in the table.
θ
0
π
6
π
4
π
3
π
2
π
6
π
4
π
3
π
2
sin θ
cos θ
3. Divide each value by 2.
θ
0
sin θ
cos θ
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4. Reduce as much as possible.
θ
0
π
6
π
4
π
3
π
2
sin θ
cos θ
Notice that the values in the table match with those values discovered when
examining the sine and cosine functions in the previous section.
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Reference Angles and Evaluating Trigonometric Functions in All Quadrants:
Using either the Unit Circle or your knowledge that sin θ = y and
cos θ = x, determine where each of the 6 trigonometric functions is
positive or negative. You should use notation such as: sin θ > 0.
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A reference angle is an angle θ0 made with the terminal side of its generating
angle θ and the nearest x-axis. Note: 0 ≤ θ0 ≤ 90◦ .
Below each of the diagrams below, write an equation for finding θ0
using θ.
Now that we can find reference angles, we may use the fact that each trigonometric function can be evaluated at a reference angle, rather than a generating
angle, while considering the sign of the function in that quadrant. This way, we
need only remember the trigonometric evaluations of functions at the angles in
the tables in the previous sections.
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