Trigonometry Lecture Notes_part1-1

Trig functions and complements as mentioned above the

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Trig Functions and Complements: As mentioned above, the two non 90 ringoperator angles must add to 90 ringoperator . This makes them complementary. This means that the cosine of one of these angles will always equal the sine of the other angle and vice versa. You will also see this pattern on the unit circle I asked you to memorize. Let’s see why, consider the drawing below:
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Consider the angles θ and B What is the sine of θ ? How does that compare with the cosine of B? (Notice that angle B = 90 ringoperator - θ ) The Cofunction Identities (note if using radian measure use 2 π instead of 90 ringoperator ) ( ) ( ) ( ) sin cos 90 tan cot 90 sec csc 90 θ θ θ θ θ θ = - = - = - ringoperator ringoperator ringoperator ( ) ( ) ( ) cos sin 90 cot tan 90 csc sec 90 θ θ θ θ θ θ = - = - = - ringoperator ringoperator ringoperator Example 12 Find a cofunction with the same value as the expression given: a. sin38 ringoperator b. csc 3 π Applications Many problems of right angle trigonometry involve the angle made with an imaginary horizontal line. The angle formed by that line and the line of sight to an object that is above the horizontal is called the angle of elevation . If the object is below the horizontal the angle is called the angle of depression .
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Example 13 A surveyor sights the top of a Giant Sequoya tree in California and measures the angle of elevation to be 32 degrees. The surveyor is standing 275 feet away from the base of the tree. What is the height of the tree? 32 ringoperator 275 Example 14 A plane must achieve a minimum angle of elevation once it reaches the end of its runway which is 200 yards from a six story building in order to clear the top of the building safely. What angle of elevation must the plane exceed in order to clear the 60 ft tall building in its path? Section 6.2 Trig Functions of Any Angle
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Let θ be any angle in standard position, and let ( ) , P x y = be a point on the terminal side of θ . If 2 2 r x y = + is the distance from ( ) 0,0 to ( ) , x y , as shown below, the six trig functions of θ are defined as: sin csc , 0 y r r y y θ θ = = cos sec , 0 x r r x x θ θ = = tan , 0 cot , 0 y x x x y y θ θ = = Example 15 Let ( ) 2,5 P = - be a point on the terminal side of θ . Find each of the six trigonometric functions of θ .
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Example 16 Find the sine function and the tangent function (if possible) at the following four quadrantal angles: a. 0 θ = b. 2 π θ = c. θ π = d. 3 2 π θ = Signs of the Trigonometric Functions Since the six trigonometric function all depend on the quantities x, y, and r and since r is always positive , the sign of a trigonometric function depends upon the quadrant in which θ lies. For example, in quadrant II we have angles of measures that are in the following set ( ) { } | 90 ,180 θ θ ringoperator ringoperator . This means that the sines of all of those angles are positive (since all y’s are positive in quad II), the cosines of all of those angles are negative (since all x’s are negative in quad II), and finally the tangents in quadrant II are always negative (since x and y have different signs).
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