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Right Triangle Trigonometry MATH 109 - Precalculus S. Rook

Right Triangle Trigonometry MATH 109 - Precalculus S. Rook

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Right Triangle Trigonometry

MATH 109 - PrecalculusS. Rook

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

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Trigonometric Functions via a Right Triangle

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:

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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:

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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

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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

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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

Trigonometric Identities

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

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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

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

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

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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

Application – Angles of Elevation & Depression

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

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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?

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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

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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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