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Section 2.1 Trig. Functions of Acute Triangles.notebook February 08, 2013 Section 2.1: Trigonometric Functions of Acute Angles Right-Triangle-Based Definition of the Trig Functions: In Chapter 1, we used angles in standard position to define the trig functions. There is another way to approach them: as ratios of the lengths of the sides of right triangles. y = side opposite angle A x = side adjacent angle A r = hypotenuse Right-Triangle-Based Definitions of Trig Functions For any acute angle A in standard position, sin A = y = opposite r hypotenuse csc A = r = hypotenuse y opposite cos A = x = adjacent r hypotenuse sec A = r = hypotenuse x adjacent tan A = y = opposite x adjacent cot A = x = adjacent y opposite Section 2.1 Trig. Functions of Acute Triangles.notebook February 08, 2013 EXAMPLE 1 Finding trigonometric function values of an acute angle Find the sine, cosine, and tangent values for angles A and B in the right triangle 8 C B 15 17 A Cofunctions In example 1, you may have noticed that sin A = cos B and cos A = sin B. Such relationships are always true for the two acute angles of a right triangle. (Note: We ALWAYS use C to be the right angle in an ABC triangle.) B a C sin A = a c = cos B tan A = a b = cot B sec A = c b = csc B c b A Section 2.1 Trig. Functions of Acute Triangles.notebook February 08, 2013 CofunctionIdentities: Foranyacuteangle A, sinA=cos(90°‐A);cosA=sin(90°‐A) cscA=sec(90°‐A);secA=csc(90°‐A) tanA=cot(90°‐A);cotA=tan(90°‐A) EXAMPLE 2 Write each function in terms of its cofunctions. (a) cos 41° (b) tan 13° (c) sec 84° EXAMPLE 3 Solving Equations Using the Cofunction Identities Find one solution for each equation. Assume all angles involved are acute angles. cos(3θ + 5°) = sin(6θ + 10°) tan(4θ - 1°) = cot(5θ + 22°) Section 2.1 Trig. Functions of Acute Triangles.notebook February 08, 2013 Comparing Function Values of Acute Triangle θ θ θ As angle θ increases from 0o to 90o, sin θ ___________. As angle θ increases from 0o to 90o, cos θ ___________. As angle θ increases from 0o to 90o, tan θ ___________. EXAMPLE 4 Tell whether each statement is true or false. (a) sin 43° > sin 22° (b) cos 71° ≤ cos 86° Section 2.1 Trig. Functions of Acute Triangles.notebook February 08, 2013 Special Right Triangles 30o - 60o - 90o θ 30o 45o 60o 45o - 45o - 90o sin θ cos θ tan θ csc θ sec θ cot θ Section 2.1 Trig. Functions of Acute Triangles.notebook February 08, 2013 Drill Quiz #2 Quadrantal and Special Right Angles!