Radian Measure That was easy What does that mean? One radian is the measure of a central angle of a circle that intercepts an arc whose length equals a radius of the circle. Central angle intercepts an arc whose length, s, is inches in a circle that has a radius length of 12 inches. Find r = 12 General Formula or or That was easy

Converting from Degrees to Radians To change from degrees to radians, multiply the number of degrees by Convert the following angle measurements from degrees to radians.

Converting from Radians to Degrees To change from radians to degrees, multiply the number of radians by Convert the following angle measurements from radians to degrees.

Comparing Degrees and Radians The radius of a circle is 12 centimeters. What is the number of centimeters in the length of the minor arc intercepted by a central angle that measures 135o? Method 1 Method 2 r = 12 s How about that. You get the same answer either way.

Angles in Standard Position Let’s look at a diagram that will explain what an angle in standard position is. y If the terminal side rotates in a counterclockwise direction, the angle is positive. Terminal Side -x x Initial Side If the terminal side rotates in a clockwise direction, the angle is negative. -y

Coterminal Angles A positive angle and a negative angle are coterminal angles if the sum of the absolute values of their degree measures is 360 or a multiple of 360. y Example 1 A rotation angle measures 210o. What is the measurement of its coterminal angle? -x x Example b A rotation angle measures 300o. What is the measurement of its coterminal angle? -y

Coordinate Trigonometric Definitions I’m not sure I like the sound of that. If P(x, y) is any point on the terminal side of an angle in standard position and r is the distance of P from the origin, then the six trigonometric functions can be defined in terms of x, y, and r. That doesn’t sound much better. y P(x, y) r y x x

Coordinate Trigonometric Example If P(4, -3) is a point on the terminal side of an angle, find the values of the sine, cosine, and tangent of the angle. Remember the Pythagorean Theorem. I think I can do this. y 4 x -3 r P(4, -3) That was easy

I think this is probably something very important. The Unit Circle A Unit Circle is a circle with center at the origin and a radius length of 1. P(x, y) r = 1 y x I think this is probably something very important.

Signs of the Trigonometric Functions The signs of the trigonometric functions depend on the signs of x and y in the quadrant in which the terminal side of the angle lies. A +y S Quadrant II Quadrant I (-x, +y) (+x, +y) Sine Positive All Positive -x +x Quadrant III Quadrant IV (-x, -y) (+x, -y) T Tangent Positive Cosine Positive C -y

Trigonometric Function Example The only quadrant where cosine is negative and tangent is positive is quadrant III. Sine is only positive in quadrants I and II -4 13 5 -3 5 -12

Reference Angles The reference angle is the acute angle whose vertex is the origin and whose sides are the terminal side of the original angle and the x-axis. The reference angle is denoted as Quadrant I Quadrant II Quadrant III Quadrant IV

Reducing Angles of Trigonometric Functions The trigonometric function of any angle can be expressed as either plus or minus the same trigonometric function of its reference angle. Let’s look at an example. Reduce cos 135o. Find the reference angle. 135o Determine the sign of the function in the quadrant in which the reference angle is located. The reference angle is in quadrant II, and in quadrant II cosine is negative. Therefore:

The 45 – 45 Right Triangle That was easy The 45 – 45 Right Triangle has two 45o angles, and two congruent sides opposite those angles. Example 1: Given one leg. 45 45 12 7 45 45 45 7 1 Example b: Given the hypotenuse. 45 45 45 4 1 45 45 4 That was easy

The 30-60-90 Right Triangle That was easy The 30-60-90 Right Triangle has a 30o angle, and a 60o angle. Example 1: Given one leg. 30 30 30 18 24 That was easy 60 60 2 9 Example b: Given the hypotenuse. 30 30 8 15 60 60 60 1 4

Trigonometric Functions of 45 Degrees Let’s take another look at the 45-45 right triangle. 45 1 45 1 That’s pretty easy, but I bet it’s really important.

Trigonometric Functions of 30 and 60 Degrees Let’s take another look at the 30-60-90 right triangle. 30 2 60 1 That’s a little more work, but it’s still pretty easy, and I’m sure it’s really important.

Summary of the Trigonometric Functions of Special Angles

Quadrantal Angles An angle rotation of 0o, 90o, 180o, 270o, or 360o is a Quadrantal Angle (x, y) 90o (0, 1) Degrees 90o 180o 270o 360o Radians r = 1 (-1, 0) (1, 0) 180o 0o, 360o 1 -1 (0, -1) -1 1 270o undefined undefined

Working with Trigonometric Functions Before we can work with trigonometric functions, one very important thing needs to be pointed out. are very different expressions. For example, if These are two very different answers because they are two very different expressions.

Operations with Trigonometric Functions When working with equations that contain trigonometric functions, it helps to treat the trigonometric function like a variable. Example 3 Example 1 Example b

More Operations with Trigonometric Functions Combining Fractions Factoring Example 1 Factoring Example b

Solving an Equation for a Trigonometric Function Hey, this stuff is really pretty easy. That was easy

Finding the Value of an Angle in an Equation with a Trigonometric Function Solve for all values of That was easy