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Overview
• Section 4.3 in the textbook:– Trigonometric functions via a right triangle– Trigonometric identities– Proving simple identities– Approximating with a calculator– Application – angles of elevation & depression
2
Trigonometric Functions via a Right Triangle
• Another way to view the six trigonometric functions is by referencing a right triangle
• You must memorize the following definition – a helpful mnemonic is SOHCAHTOA:
4
hypotenuse
oppositesine
hypotenuse
adjacentcosine
adjacent
oppositetangent
opposite
hypotenuse
sine
1cosecant
adjacent
hypotenuse
cosine
1secant
opposite
adjacent
tangent
1cotangent
Trigonometric Functions via a Right Triangle (Example)
Ex 1: Use the diagram and find the value of the six trigonometric functions of θ:
5
Special Triangles – 30° - 60° - 90° Triangle
• Think about taking half of an equilateral triangle– Shortest side is x and is opposite the 30° angle– Medium side is and is opposite the 60°
angle– Longest side is 2x and is
opposite the 90° angle
3x
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Special Triangles – 45° - 45° - 90°
• Think about taking half of a square along its diagonal– Shortest sides are x and are opposite the 45°
angles– Longest side is and is
opposite the 90° angle2x
7
Cofunctions
• The six trigonometric functions can be separated into three groups of two based on the prefix co:– sine and cosine– secant and cosecant– tangent and cotangent
• Each of the groups are known as cofunctions• The prefix co means complement or opposite
8
Cofunction Theorem
• Cofunction Theorem: If angles A and B are complements of each other, then the value of a trigonometric function using angle A will be equivalent to its cofunction using angle B or vice versa
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ABBA
ABBA
ABBA
BA
cottan AND cottan
cscsec AND cscsec
cossin AND cossin
:90 If
Special Triangles (Example)
Ex 2: Use an appropriate special triangle to find the following:
a) sin 45°, cos 45°, tan 45°b) sin 30°, cos 30°, tan 30°c) sin π⁄3, cos π⁄3, tan π⁄3
d) sec 45°, csc 30°, cot π⁄3
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Trigonometric Functions of Common Angles
Degrees Radians cos θ sin θ
0° 0 1 0
30°
45°
60°
90° 0 1
180° -1 0
270° 0 -1
360° 1 0
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6
4
3
2
2
3
2
2
3
2
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2
2
1
2
2
2
1
2
1
2
1
Reciprocal Identities
• The following are the reciprocal identities which you must MEMORIZE:
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sin
1csc
csc
1sin
tan
1cot
cos
1sec
sec
1cos
cot
1tan
Ratio Identities
• Allows us to write tangent and cotangent in terms of sine and cosine:
• Again, you must MEMORIZE these identities• Can verify using our definitions in terms of the
unit circle:
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cos
sintan
sin
coscot
x
y
cos
sintan
y
x
sin
coscot
Pythagorean Identities
• Important identities that make solving certain types of trigonometric problems easier:
• You must MEMORIZE at least the first identity • sin2θ is equivalent to (sin θ)2
15
1cossin 22
22 sectan1
22 csccot1
Trigonometric Identities
Ex 3: Use the given information to solve:
a) Given sin θ = ¼, find the exact value of cos θ and then cot θ
b) Given , find cot α and sec(90° – α)
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3
4csc
Proving Simple Identities
• Objective is to transform the left side into the right side one step at a time by using:– Multiplication– Addition/Subtraction– Identities
• It takes CONSIDERABLE PRACTICE to fully understand the process of proving identities
• We will be proving more complex identities later in the course so be sure to understand how to prove the simpler identities!
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Proving Simple Identities (Example)
Ex 4: Use Trigonometric identities to transform the left side of the equation into the right side:
a)
b)
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1tansectansec
seccscsin
cos
cos
sin
Approximating with a Calculator
• ESSENTIAL to know when to use degree mode and when to use radian mode:– Angle measurements in degrees are post-fixed
with the degree symbol (°)– Angle measurements in radians are sometimes
given the post-fix unit rad but more commonly are given with no units at all
21
Approximating with a Calculator
Ex 5: Approximate the following with a calculator – make sure to use the correct units:
a) sin 85°
b) sec 11° 59’
c) cot 322
Angle of Elevation and Angle of Depression
• Angle of Elevation: angle measured from the horizontal (or flat line) upwards
• Angle of Depression: angle measured from the horizontal (or flat line) downwards
24
Angle of Elevation (Example)
Ex 6: A ladder is leaning against the top of a 20-foot wall. If the angle of elevation from the ground to the ladder is 37°, what is the length of the ladder?
25
Angle of Depression (Example)
Ex 7: A person standing on the roof of a building notices that he has an angle of depression of 15° with a landmark on the ground. If the distance from the building to the landmark is 100 feet, approximately how tall is the building
26
Summary
• After studying these slides, you should be able to:– Express the six trigonometric functions in terms of the sides of
a right triangle– State and use important trigonometric identities– Prove simple properties– Use a calculator to approximate trigonometric functions– Use angles of elevation & depression to solve problems
• Additional Practice– See the list of suggested problems for 4.3
• Next lesson– Trigonometric Functions of Any Angle (Section 4.4)
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