5.6 Angles and Radians (#1,2(a,c,e),5,7,15,17,21) 10/5/20151

Radian Measure To talk about trigonometric functions, it is helpful to move to a different system of angle measure, called radian measure. A radian is the measure of a central angle whose intercepted arc is equal in length to the radius of the circle. 10/5/20152

Radian Measure There are 2  radians in a full rotation -- once around the circle There are 360° in a full rotation To convert from degrees to radians or radians to degrees, use the proportion 10/5/20153

Sample Problems Find the degree measure equivalent of radians. 10/5/20154  Find the radian measure equivalent of 210°

5.7 Solving trigonometric Equatioins 10/5/2015 (# 5.7 Solving Trigonometric Equations (#3(a,b,c,e,g),4,8,9(a,b,c,d,e,h,i),11)

Trigonometric Identities An identity is an equation which is true for all values of the variable. There are many trig identities that are useful in changing the appearance of an expression. The most important ones should be committed to memory. 10/5/20156

Trigonometric Identities Reciprocal Identities 10/5/20157  Quotient Identities

Trigonometric Identities Cofunction Identities – The function of an angle = the cofunction of its complement. 10/5/20158

Trigonometric Identities  Pythagorean Identities  The fundamental Pythagorean identity 10/5/20159  Divide the first by sin 2 x  Divide the first by cos 2 x

Solving Trig Equations Solve trigonometric equations by following these steps: – If there is more than one trig function, use identities to simplify – Let a variable represent the remaining function – Solve the equation for this new variable – Reinsert the trig function – Determine the argument which will produce the desired value 10/5/201510

Solving Trig Equations To solving trig equations: – Use identities to simplify – Let variable = trig function – Solve for new variable – Reinsert the trig function – Determine the argument 10/5/201511

Sample Problem Solve – Use the Pythagorean identity (cos 2 x = 1 - sin 2 x) – Distribute – Combine like terms – Order terms 10/5/201512

Sample Problem Solve – Use the Pythagorean identity (cos 2 x = 1 - sin 2 x) – Distribute – Combine like terms – Order terms 10/5/201513

Sample Problem Solve 10/5/  Let t = sin x  Factor and solve.

Sample Problem Solve 10/5/  Replace t = sin x.  t = sin(x) = ½ when  t = sin(x) = 1 when  So the solutions are