Trigonometry Lecture Notes_part1-1

This leads us to the idea that 360 2 π we can then

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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 ringoperator b) 135 - ringoperator 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 =
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We should memorize these diagrams
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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 ringoperator ) angles. Consider the diagram below:
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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:
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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 ringoperator angle ( 4 π ) below:
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Example 9 Evaluate sine, cosine, and tangent functions for the 60 3 or π ringoperator and the 30 6 or π ringoperator 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:
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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 =
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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. The Pythagorean Identities 2 2 sin cos 1 θ θ + = 2 2 1 tan sec θ θ + = 2 2 1 cot csc θ θ + = Example 11 Given that 3 sin 5 θ = and that θ is an acute angle, find the value of cos θ using the Pythagorean identity.
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