Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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 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: opposite sine hypotenuse 1 hypotenuse cosecant sine opposite adjacent cosine hypotenuse 1 hypotenuse secant cosine adjacent opposite tangent adjacent 1 adjacent cotangent tangent opposite 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) opp A y hyp r sin A csc A hyp r opp A y cos A adj A x hyp r sec A hyp r adj A x adj A x opp A y cot A tan A opp A y adj A x 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 opp A y adj B sin A cos B hyp r hyp adj A x opp B cos A sin B hyp r hyp hyp r hyp sec A csc B adj A x opp B hyp r hyp csc A sec B opp A y adj B opp A y adj B tan A cot B adj A x opp B adj A x opp B cot A tan B opp A y adj B 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 sin A cos B AND sin B cos A If A B 90 : sec A csc B AND sec B csc A tan A cot B AND tan B cot A 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 Exact Values for Common Angles • For select angles, we can obtain exact values for the trigonometric functions: x x 3 3 cos 30 2x 2 x cos 60 1 2 sin 45 2 x 2 2 1 2 cos 45 2 x 2 2 x 1 2x 2 sin 30 x 1 2x 2 sin 60 x 3 3 2x 2 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. 3 sin 60 tan 60 2 3 1 cos 60 2 • Also: cos 0 1 cos 90 0 sin 0 0 sin 90 1 sin 90 1 tan 90 undefined cos 90 0 15 Exact Values for Common Angles (Continued) • In summary, this chart MUST be memorized by chapter 3: θ cos θ sin θ 0° 1 0 30° 3 2 1 2 1 2 2 2 1 2 2 2 60° 1 2 3 2 90° 0 1 45° 16 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