Definition II: Right Triangle Trigonometry Trigonometry MATH 103 S. Rook

Overview Section 2.1 in the textbook: – Right triangle Trigonometry – Cofunction theorem – Exact values for common angles 2

Right Triangle Trigonometry

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

Right Triangle Trigonometry (Continued) Definition II is an extension of Definition I as long as angle A is acute (why?): – Lay the right triangle on the Cartesian Plane such that A is the origin and B is the point (x, y) 5

Right Triangle Trigonometry (Example) Ex 1: For each right triangle ABC, find sin A, csc A, tan A, cos B, sec B, and cot B: a) b)C = 90°, a = 3, b = 4 6

Cofunction Theorem

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

Cofunctions and Right Triangles 9

Cofunctions and Right Triangles (Continued) The measure of the angles in a triangle must sum to 180° By definition, a right triangle contains a right angle measuring 90° (C = 90°) Therefore, the remaining two angles must sum to 90° (A + B = 90°) 10

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 11

Cofunction Theorem (Example) Ex 2: Use the Cofunction Theorem to fill in the blanks so that each equation becomes a true statement: a)cot 12° = tan ____ b)sec 39° = csc ____ c)sin 80° = ___ 10° 12

Exact Values for Common Angles

For select angles, we can obtain exact values for the trigonometric functions: 14

Exact Values for Common Angles (Continued) Only need to memorize the sine and cosine values: – Can derive the remaining trigonometric functions through identities e.g. Also: 15

Exact Values for Common Angles (Continued) In summary, this chart MUST be memorized by chapter 3: 16 θcos θsin θ 0°10 30° 45° 60° 90°01

Exact Values for Common Angles (Example) Ex 3: For each of the following, replace x with 30°, y with 45°, and z with 60°, and then simplify as much as possible: a)3sin(2y) b)2sec(90° – z) c)4csc(x) 17

Summary After studying these slides, you should be able to: – Apply the six trigonometric functions to a right triangle – State the definition of a cofunction – Understand and use the Cofunction Theorem – State and use values of the trigonometric functions for common angles Additional Practice – See the list of suggested problems for 2.1 Next lesson – Calculators and Trigonometric Functions of an Acute Angle (Section 2.2) 18