Trigonometry Lecture Notes_part1-1

Example 17 if tan θ 0 and cos θ 0 name the quadrant

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Example 17 If tan θ > 0 and cos θ < 0, name the quadrant in which angle θ lies. Example 18 Given 2 tan 3 θ - = and cos 0 θ > , find cos θ and csc θ . Reference Angles We will often evaluate the trigonometric functions of positive angles greater than 90 ringoperator and all negative angles by making use of a positive acute angle. This angle is called a reference angle . Definition of a Reference Angle Let θ be a non-acute angle in standard position that lies in a quadrant. Its reference angle is the positive acute angle θ ’ formed by the terminal side of θ and the x-axis.
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If 90 180 θ < < ringoperator ringoperator , then 180 β θ = - ringoperator If 180 270 θ < < ringoperator ringoperator , then 180 β θ = - ringoperator If 270 360 θ < < ringoperator ringoperator , then 360 β θ = - ringoperator Example 19 Find the reference angle, β , for each of the following angles: a. 345 θ = ringoperator b. 5 6 π θ = c. 135 θ = - ringoperator d. 2.5 θ = Using Reference Angles to Evaluate Trigonometric Functions The value of a trigonometric function of any angle θ is found as follows: 1. Find the appropriate reference angle, β . 2. Determine the required function value for β (i.e.-sine, cos, tan, …). 3. Use the quadrant that θ lies in to determine the appropriate sign for the answer from step 2. Example 20 Use a reference angle to determine the exact value of each of the following trig functions: a. sin135 ringoperator b. 4 cos 3 π c. cot 3 π - Section 6.3 Trigonometric Functions of Real Numbers; Periodic Functions
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The unit circle we have been working with has the equation 2 2 1 x y + = , if we were to solve this equation of a circle for y this equation does not form a function of x (clearly, it would fail the vertical line test). However, recall our arc length formula s r θ = , for the unit circle, r is equal to one. This means our arc length formula will give us this simpler relationship: 1 s θ = (or just s θ = ). The length of the intercepted arc is equal to the radian measure of the angle. Now let’s change up the notation a bit. We will call the angle t instead of θ in this section. Now consider the diagram below: For each real number value for t , there is a corresponding point P = (x,y) on the unit circle. We can now define the cosine function at t to be the x-coordinate of P and the sine function at t to be the y- coordinate of P. Namely: cos x t = and sin y t = . The Definitions of the Trigonometric Functions in Terms of a Unit Circle If t is a real number and P = (x, y ) is a point on the unit circle that corresponds to t , then: sin t y = cos t x = tan , 0 y t x x = 1 csc , 0 t y y = 1 sec , 0 t x x = cot , 0 x t y y = Example 21Use the figure to find the values of the trig functions at t = 8 π
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Domain and Range of the Sine and Cosine Functions Let’s consider the unit circle definitions of sine and cosine, namely: cos t x = and sin t y = . Since t corresponds to an arc length along the unit circle it can be any real number. This means that the domain for these two functions must be all real numbers. Now, because the radius of the unit circle is one, we have restrictions on the values that are possible for both y and x, so the ranges will be given by: [ ] 1,1 - . That is 1 cos 1 t - ≤ , and 1 sin 1 t - ≤
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