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Math 4 Pre-Calculus Name________________________ Date_________________________ Trigonometric Functions of Right Triangles — 6.2 Definitions A triangle is a right triangle if one of its angles is a right angle. See pictures below. The sine of θ , denoted s i n θ , is the ratio of the side opposite θ to the hypotenuse. The cosine of θ , denoted c o s θ , is the ratio of the side adjacent θ to the hypotenuse. The tangent of θ , denoted t a n θ , is the ratio of the side opposite θ to the side adjacent θ . In summary: sinθ = opp h yp co sθ = adj h yp You can remember this by: SOH CAH TOA 1. Find sine, cosine and tangent for the angle θ . 2. If θ is an acute angle and s i n θ = 3 , find c o s θ and t a n θ . 4 tanθ = opp adj Reciprocal Trigonometric Functions The cosecant of θ , denoted c s c θ , is the reciprocal of the sine θ or the ratio of the hypotenuse to the side opposite θ . The secant of θ , denoted s e c θ , is the reciprocal of the cosine θ or the ratio of the hypotenuse to the side adjacent θ . The cotangent of θ , denoted c o t θ , is the reciprocal of the tangent θ or the ratio of the side adjacent θ to the side opposite θ . In summary: cscθ = 1 h yp = sinθ opp 3. Find all six trigonometric functions for the angle θ . s e cθ = 1 hyp = cosθ adj cotθ = 1 adj = tanθ opp θ y x 4. Label all of the angles of the two triangles both in degrees and radians, and then label the lengths of each side. Isosceles Right Triangle 30-60-90 Triangle Use the isosceles right triangle and the 30-60-90 triangle above to fill in the chart below. θ (radians) θ (degrees) 30° π 4 60° sinθ cosθ tanθ cotθ s e cθ cscθ 5. Find exact values of x and y . 6. Distance to Mt. Fuji The peak of Mt. Fuji in Japan is approximately 12,400 feet high. A trigonometry student, several miles away notes that the angle between level ground and the peak is 30°. Estimate the distance from the student to the point on level ground directly beneath the peak. 7. Approximate to four decimal places, when appropriate. a) sin 75° b) cot π 5 Fundamental Trigonometric Identities Reciprocal Identities: 1 1 cscθ = secθ = sinθ cosθ Tangent and Cotangent Identities: sinθ tanθ = co sθ Pythagorean Identities: s i n 2 θ + c o s2 θ = 1 sinθ . co sθ 8. Show t a n θ = 9. Show s i n 2 θ + c o s 2 θ = 1 cotθ = cotθ = 1 tanθ cosθ sinθ 1 + t a n 2 θ = s e c2 θ 1 + c o t 2 θ = c s c2 θ 10. Use the Pythagorean identities to write the expression as an integer. ( 2 ) − 3cot (β 2 ) 3 c s c2 β 11. 2 Simplify the expression. ( c s cθ + 1 1 s i n2 θ ) + cscθ 12. Use the fundamental identities to write the first expression in terms of the second, for any acute angle θ . cosθ , sinθ 13. Verify the identity by transforming the left-hand side into the right-hand side. tanθ cotθ = 1 14. Verify the identity by transforming the left-hand side into the right-hand side. c o s 2 ( 2θ ) − s i n 2 ( 2θ ) = 2 c o s 2 ( 2θ ) − 1 15. Verify the identity by transforming the left-hand side into the right-hand side. co t θ + t a n θ = c s cθ s e cθ