Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Section 2.4: Trigonometric Identities It will be important to know these now as we will work on them down the road. First, some reminders: What we will add: Pythagorean Identities Some definitions first. In class derivation: We can “divide out” sine or cosine to get two more identities. In Class Example (see example 3 on pg 114): In Class Example (see example 5 on pg 115): Section 3.1: Trigonometric Identities Instead of using degrees, we can also look at angles using a different system. Let’s define some stuff. A central angle is an angle that has its vertex at the center of a circle. So the technical definition of a radian is the length of the arc on the circle divided by the length of the radius. Here are some key angles that lead to some key radians. When the length of the arc on the circle, (aka “s”) equals the length of the radius, (aka “r”), s/r becomes 1. A radian is about 57°. The length of the entire circle (technically the circumference) depends on the radius. A circle with radius r will have a circumference of 2πr. So the radian measure of all 360° of a circle would be: 360° in radians = ௦ = ଶగ = 2ߨ In Class Example (see examples 1 and 2 on pg 131): What is the measure (in radians) of a central angle ϴ that intercepts an arc of length 12 millimeters on a circle with radius 4 centimeters? A List of the Metric Prefixes Prefix Symbol Numerical kilo k 1,000 hecto h 100 deca da 10 no prefix: 1 100 deci d 0.1 centi c 0.01 milli m 0.001 Because radians are unitless, the word radians (or rad) is often omitted. If an angle measure is given simply as a real number, then radians are implied. WORDS The measure of ϴ is 4 degrees. → ϴ = 4° The measure of ϴ is 4 radians. → ϴ = 4 Converting Between Degrees and Radians Recall: 360° in radians ௦ = ଶగ = 2ߨ If 360°=2 π, then it stands to reason that 180°= π and 90°= π/2 and 45°= π/4 Notice: 180°= π gives us a way to convert between radians and degrees. In Class Example (see examples 3-6 on pgs 133-134): a) Convert 460° to radians. b) Convert 7 radians to degrees. Use the π key! Reference Page In Class Example (see examples 7-9 on pgs 135-137): Find the reference angle for ହగ ଷ and the exact value of cosine. Show all work. Rationalize denominators where needed. Section 2.4: 8, 16, 17, 21, 22, 27, 32, 35, 39, 42, 57 Problem hints: 2.4.32: find cosine first to calculate tangent. 2.4.39 and 2.4.42: set up triangle in the right quadrant and calculate hypotenuse Section 3.1: 7, 8, 20, 21, 34, 35, 42, 43, 48, 49, 54, 55, 68*, 74*, 78*, 80*, 92♦ *For these problems, state the reference angle and quadrant as well. ♦ For this problem, they’re basically saying that 45 minutes is equal to 2π. So what fraction of 2π would 25 minutes give you? Show work and fractions.