Right Triangle Trigonometry 4.3

A standard right triangle hypotenuse Side opposite of θ θ Side adjacent to θ

Right Triangle Definitions of Trigonometric Functions Let θ be an acute angle of a right triangle. Then the six trigonometric functions of the angle θ are defined as follows: sin θ = csc θ= cos θ = sec θ = tan θ = cot θ =

Example 1: Evaluating Trig. Functions Use the triangle to find the exact values of the six trigonometric functions of θ. 4 θ 3

Example 2: Evaluating Trig Functions of 45° Find the exact values of sin 45°, cos 45°, and tan 45°.

Example 3: Evaluating Trig. Functions of 30° and 60° Us the 30/60/90 triangle to find the exact values of sin 60°, cos 60°, sin 30°, cos 30°. 2 60° 1

Sines, Cosines, and Tangents of Special Angles sin 30° = sin = cos 30° = cos = tan 30° = tan = sin 60° = sin = cos 60° = cos = tan 60° = tan = sin 45° = sin = cos 45° = cos = tan 45° = tan = 1

NOTE: Sin 30° = ½ = cos 60°. This is true because 30° and 60° are complementary angles. So, cofunctions of complementary angles are equal. sin (90° - θ) = cos θ cos (90° - θ) = sin θ tan (90° - θ) = cot θ cot (90° - θ) = tan θ sec (90° - θ) = csc θ csc (90° - θ) = sec θ

Fundamental Trigonometric Identities Reciprocal Identities Sin θ = csc θ = Cos θ = sec θ = Tan θ = cot θ = Quotient Identities tan θ = cot θ = Pythagorean Identities sin2θ + cos2θ = 1 1 + tan2θ = sec2θ 1 + cot2θ = csc2θ

Example 4: Applying Trig. Identities Let θ be an acute angle such that sin θ = .6. Find the values of A) cos θ and B) tan θ using trigonometric identities.

Example 5: Using Trig. Identities Use trigonometric identities to transform one side of the equation into the other. (Prove the statement is true!) cos θ∙sec θ = 1 (sec θ + tan θ)(sec θ – tan θ) = 1