Trigonometric Functions Unit 1 Lesson 5 Trigonometric Functions

Before we get started with trig… When you were younger, you learned how to state the measure of an angle in degrees. You learned that a circle was formed by rotating the terminal side of an angle 360° Why “360”? (A little Babylonian history here…)

Before we get started with trig… As you matured mathematically, we, as teachers, threw a curveball at you, and said “You’re no longer going to work in degrees, but, instead, you are now going to work in radians! You now learned that a circle was formed by rotating the terminal side of an angle by 2𝜋. Why “2𝜋”? (A little algebra here…) 𝜋= 𝐶 𝑑 𝜋= 𝐶 2𝑟 𝐶=2𝜋𝑟 For a circle with 𝑟=1 𝐶=2𝜋 There are 2  radians in every circle, which is  6.8 radians. 1 rad ≈57.3°

Degree measures and radian measures have the following relationship: To help you make the switch from degrees to radians and vice-versa, we showed you this … Degree measures and radian measures have the following relationship: 2 radians = 36 0 ° and 1 radians = 180° We can use this relationship to convert from degrees to radians and vice-versa! Convert degree to radian: multiply degrees by 𝜋 𝑟𝑎𝑑 180 𝑑𝑒𝑔 Convert radian to degree: multiply radians by 180 𝑑𝑒𝑔 𝜋 𝑟𝑎𝑑

Convert radian to degree Remember, if the value of an angle does not have the degrees symbol, then it is assumed to be in radians! In AP Calculus, we work entirely in radians! A. Convert 𝝅 𝟐 B. Convert 2 2 𝑟𝑎𝑑 180 𝑑𝑒𝑔 𝜋 𝑟𝑎𝑑 𝜋 2 𝑟𝑎𝑑 180 𝑑𝑒𝑔 𝜋 𝑟𝑎𝑑 = 360 𝑑𝑒𝑔 𝜋 ≈114.59° =90°

Convert degree to radians a. 135 ° b. -270 ° (135°) 𝜋 𝑟𝑎𝑑 180° (−270°) 𝜋 𝑟𝑎𝑑 180° = 3𝜋 4 𝑟𝑎𝑑 =− 3𝜋 2 𝑟𝑎𝑑 Again, If neither degrees or radians are stated, then RADIANS is the default ( 3𝜋 4 = 3𝜋 4 𝑟𝑎𝑑𝑖𝑎𝑛𝑠)

We also taught you that … Positive angle: The measure of an angle is positive if a ray is rotated counterclockwise, starting from the right side of the x-axis. Negative angle: The measure of an angle is negative if a ray is rotated clockwise.

We showed you that there was a variety of ways to get to the same spot on the unit circle. coterminal angles: Two angles that have the same initial and terminal sides. Angles are often named with Greek letters: = alpha = beta = theta (used the most) 𝜑= phi You can loop around to the same spot an infinite number of times by adding or subtracting multiples of 2𝜋 to a radian value.

We merged the idea of right triangle trigonometry with the unit circle By drawing special right triangles within the unit circle, we are able to compute the cosine and sine values for special points on the unit circle. cos 𝑥 , sin 𝑥

You were asked to memorize the values of cosine and sine of these special angles.

Some teachers even like to show how the graphs of sine and cosine relate to the unit circle!

Knowing the graphs of sine and cosine really do help! Did ya know that the graph of cosine is just a shift left of the graph of sine by 𝜋 2 units?

Speaking of sine and cosine, it is good to know their parity (odd/even)! Looking at the graph of sine, we see that it is an odd function (symmetric about the origin, rotated 180°). Sine is periodic, repeating its pattern and values every 2𝜋 units. sin 𝑥 = sin 𝑥+2𝜋𝑘 Since it is an odd function, we can say sin −𝜃 =− sin 𝜃

Speaking of sine and cosine, it is good to know their parity (odd or even)! Looking at the graph of cosine, we see that it is an even function (symmetrical to the y-axis). Cosine is periodic, repeating its pattern and values every 2𝜋. cos 𝑥 = cos 𝑥+2𝜋𝑘 Since it is an even function, we can say cos −𝜃 = cos 𝜃

Introducing the 4 other trig functions We know from our last lesson that the 4 other trig functions can be created using sine and cosine Since these 4 functions are rational functions, we know that they all have vertical asymptotes! sec 𝑥 = 1 cos 𝑥 csc 𝑥 = 1 s𝑖𝑛 𝑥 tan 𝑥 = sin 𝑥 cos 𝑥 co𝑡 𝑥 = cos 𝑥 sin 𝑥

Tangent and Cotangent graphs Both tan and cot repeat their patterns every 𝜋 𝑢𝑛𝑖𝑡𝑠

Secant and Cosecant graphs Both sec and csc repeat their patterns every 2𝜋 𝑢𝑛𝑖𝑡𝑠

Computing trig values We can use our knowledge of the sine and cosine values in the 1st quadrant to help us compute the values of any trig values at other “related” angle values.

Trig Identities sin 2 𝑥 + cos 2 𝑥 =1 Basic tan 2 𝑥 +1= sec 2 𝑥 There are many, many trig identities! Pythagorean Identities (3 of them): sin 2 𝑥 + cos 2 𝑥 =1 Basic tan 2 𝑥 +1= sec 2 𝑥 1+ cot 2 𝑥 = csc 2 𝑥 The last 2 identities can be obtained by dividing the top “basic” identity by either cos 2 𝑥 or sin 2 𝑥 .

Trig Identities

A Tricky problem (for some of you) Suppose that cos 𝜃 = 2 5 . Calculate tan 𝜃 in the following two cases: A) 0<𝜃< 𝜋 2 and B) 𝜋<𝜃<2𝜋 What would be a smart thing to do? Draw a right triangle in the 1st quadrant, label sides, and use the Pythagorean Theorem to figure out the 3rd side! cos 𝜃 = 𝑎𝑑𝑗 ℎ𝑦𝑝 = 2 5 𝑜𝑝𝑝= 5 2 − 2 2 = 21

A Tricky problem (for some of you) Suppose that cos 𝜃 = 2 5 . Calculate tan 𝜃 in the following two cases: A) 0<𝜃< 𝜋 2 Since 𝜃 is in the 1st quadrant, we know that both sine and cosine have positive values! tan 𝜃 = sin 𝜃 cos 𝜃 = 21 5 2 5 = 21 2

A Tricky problem (for some of you) Suppose that cos 𝜃 = 2 5 . Calculate tan 𝜃 in the following two cases: B) 𝜋<𝜃<2𝜋 Since cos 𝜃 is positive, we know that the triangle is in the 4th quadrant, meaning that sin 𝜃 is going to be negative! tan 𝜃 = sin 𝜃 cos 𝜃 = − 21 5 2 5 = − 21 2

Homework Chapter 1 Precalculus Review Packet Pgs. 9 - 10, #1, 3, 4, 6, 7 – 15 odd, 16, 19 – 23 odd