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Unformatted text preview: This leads us to the idea that 360 ° = 2 π We can then say that 180 π ° = , and finally we can use a convenient form of one, 1 = 180 π ° Example 5 Convert the following into radians: a) 60 r b) 135 r Example 6 Convert the following into degrees: a) 3 π radians b) 5 3 π radian c) 1 radian The Length of a Circular Arc Recall that we said earlier that length of the intercepted arc s radius r θ = = , well this leads to the following formula: s r θ = We should memorize these diagrams Section 6.2 Right Angle Trigonometry Triangles have three angles. Some special triangles have a 90 degree angle (just one of course, since the sum of all angles in a triangle must add to 180 degrees). We want to be able to talk about some special names for the sides of these triangles relative to one of the other (non 90 r ) angles. Consider the diagram below: Now we can talk about the six trigonometric functions. These functions are like the functions you know of already, such as ( ) f X , but the six trig functions have names instead of the single letter names of f, g, and h. Here are the names and their abbreviations: Name Abbreviation Name Abbreviation Sine Sin Cosecant Csc Cosine Cos Secant Sec Tangent Tan Cotangent Cot Here is how we will define these functions: Notice the functions in the left column (sine, cosine, and tangent) are reciprocals of the right column. [Memory aids: Soh, Cah, Toa, or my favorite S: O scar H ad, C: A H eap, T: O f A pples] Example 7 Assume b = 12 and a = 5, then find the value of each of the six trig functions of angle A. Special Angles Example 8 Evaluate the six trigonometric functions for the 45 r angle ( 4 π ) below: Example 9 Evaluate sine, cosine, and tangent functions for the 60 3 or π r and the 30 6 or π r below: I will require you to memorize the sine and cosine values for the 16 angles shown below on the Unit Circle : Below is another version that is computer drawn: Fundamental Identities The reciprocal identities : 1 sin csc 1 csc sin θ θ θ θ = = 1 cos s c 1 sec cos e θ θ θ θ = = 1 tan cot 1 cot tan θ θ θ θ = = The quotient identities : sin tan cos θ θ θ = cos cot sin θ θ θ = Example 10 Given 1 sin 2 θ = and 3 cos 2 θ = , find the value of each of the remaining trigonometric functions. Recall the Pythagorean Theorem for right triangles, 2 2 2 a b c + = , and consider the following triangle: Let’s divide the equation by 2 c , then we have 2 2 2 2 1 a b c c + = Which is the same as 2 2 1 a b c c + = Relative to θ in our drawing this leads us to perhaps the most important the identity in this course: 2 2 sin cos 1 θ θ + = ***Please note that ( ) 2 2 sin sin θ θ = By using the same approach but dividing by either 2 a and 2 b , we can arrive at the following identities....
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 Fall '12
 lipsh
 Trigonometry, Cos, Inverse function, Inverse trigonometric functions

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