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**Right Triangle Trigonometry**

4.3

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**A standard right triangle**

hypotenuse Side opposite of θ θ Side adjacent to θ

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**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 =

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**Example 1: Evaluating Trig. Functions**

Use the triangle to find the exact values of the six trigonometric functions of θ. 4 θ 3

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**Example 2: Evaluating Trig Functions of 45°**

Find the exact values of sin 45°, cos 45°, and tan 45°.

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**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

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**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

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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

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**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 = tan2θ = sec2θ 1 + cot2θ = csc2θ

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**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.

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**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

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