Right Triangle Trigonometry MATH Precalculus S. 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 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

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 6

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° angle 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 9

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 10

Trigonometric Functions of Common Angles DegreesRadianscos θsin θ 0°010 30° 45° 60° 90°01 180°0 270°0 360°10 11

Trigonometric Identities

Reciprocal Identities The following are the reciprocal identities which you must MEMORIZE: 13

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: 14

Pythagorean Identities Important identities that make solving certain types of trigonometric problems easier: You must MEMORIZE at least the first identity sin 2 θ is equivalent to (sin θ) 2 15

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) 16

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! 18

Proving Simple Identities (Example) Ex 4: Use Trigonometric identities to transform the left side of the equation into the right side: a) b) 19

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 3 22

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 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) 27