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Trigonometric Functions 2.2 – Definition 2

This resource provides an overview of trigonometric functions and practical applications, illustrated through examples such as determining the length of a ski run using the angle of elevation, as well as navigation problems involving plane bearings. It explains how to define trigonometric functions on the coordinate plane and calculate their values for various angles, including acute, nonacute, and quadrant angles. By working through examples, learners will strengthen their understanding of essential trigonometric principles.

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Trigonometric Functions 2.2 – Definition 2

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  1. Trigonometric Functions2.2 – Definition 2 JMerrill, 2006 Revised, 2009 (contributions from DDillon)

  2. Angle of Elevation Review • The angle of elevation of from the ground to the top of a mountain is 68o. If a skier at the top of the mountain is at an elevation of 4,200 feet, how long is the ski run from the top to the base of the mountain? • 4,529.85 feet

  3. Navigation Review • If a plane takes off on a heading of N 33o W and flies 12 miles, then makes a right (90o) turn, and flies 9 more miles, what bearing will the air traffic controller use to locate the plane? How far is the plane from where it started? • The plan is 15 miles away on a bearing of N 8.41o E

  4. Defining Trig Functions Let there be a point P (x, y) on a coordinate plane. P(x, y) y r r is the distance from the origin to the point P, which can be represented as being on the terminal side of θ. Since r represents a distance, it is always positive and cannot = 0. The six trigonometric functions are: θ x

  5. Definition 2 Remember, the denominator cannot ever = 0!

  6. Calculating Trig Values for Acute Angles If the terminal side of θ in standard position passes through point P (6, 8), draw θ and find the exact value of the six trig functions of θ. P(6, 8) r = 10 8 r is the hypotenuse and can be found using Pythagorean Thm: x2 + y2 = r2 θ 6

  7. You Do • If the terminal side of θ in standard position passes through point P (3, 7), draw θ and find the exact value of the six trig functions of θ.

  8. Calculating Trig Values for Nonacute Angles If the terminal side of θ in standard position passes through point P (-4, 2), draw θ and find the exact value of the six trig functions of θ. P(-4, 2) r = 2√5 2 θ -4

  9. Calculating Trig Values for Quadrant Angles • Find the exact value of the six trig functions when θ=90o • A convenient point on the terminal side of 90ois (0,1). So x = 0, y = 1, r = 1 (0,1) Now, if the angle is 180o, what point will you use?

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