5.3 and 5.4 Evaluating Trig Ratios for Angles between 0 and 360

5.3 and 5.4 Evaluating Trig Ratios for Angles between 0 and 360 What has to be broken before it can be used?

5.3 and 5.4 Understanding Angles Greater than y Terminal Arm We're going to explore how triangles in a Cartesian plane have trig ratios that relate to each other θ x Initial Arm

Angles, Angles, Angles • An angle is formed when a ray is rotated about a fixed point called the vertex Terminal Arm (the part that is rotated) Vertex θ Initial Arm (does not move)

Think of Hollywood Terminal Arm Depending on how hard the director wants to snap the device, he/she will vary the angle between the initial arm and the terminal arm Initial Arm

y x • The trigonometric ratios have been defined in terms of sides and acute angles of right triangles. • Trigonometric ratios can also be defined for angles in standard position on a coordinate grid. Coordinate grid

Standard Position • An angle is in standard position if the vertex of the angle is at the origin and the initial arm lies along the positive x-axis. The terminal arm can be anywhere on the arc of rotation y Greek Letters such as α,β,γ,δ,θ (alpha, beta, gamma, delta, theta) are often used to define angles! Terminal Arm θ x Initial Arm

Terminal Arm Initial arm Standard Form For example…… Terminal Arm Initial arm Not Standard Form Terminal Arm Initial arm

Positive and Negative angles Positive angles Negative angles θ θ A positive angle is formed by a counterclockwiserotation of the terminal arm A negative angle is formed by a clockwiserotation of the terminal arm

The Four Quadrants The x-y plane is divided into four quadrants. If angle θ is a positive angle, then the terminal arm lies in which quadrant? Quadrant I Quadrant II 0º< θ < 90º Quadrant III Quadrant IV 90º < θ < 180º 180º < θ < 270º 270 º < θ < 360º

Principal Angle and Related Acute Angle The principal angle is the angle between the initial arm and the terminal arm of an angle in standard position. Its angle is between 0º and 360º The related acute angle is the acute angle between the terminal arm of an angle in standard position (when in quadrants 2, 3, or 4).and the x-axis. The related acute angle is always positive and is between 0º and 90º Terminal Arm θ Principal Angle β Related Acute Angle Initial Arm Let’s look at a few examples……

In these examples, θ represents the principal angle and β represents the related acute angle θ θ β Principal angle: 65º Principal angle: 140 º Related acute angle: 40º No related acute angle because the principal angle is in quadrant 1 θ θ β Principal angle: 225º Related acute angle: 45º β Principal angle: 320º Related acute angle: 40º

Notice anything? • In the first quadrant the principal angle and related acute angle are always the same • In the second quadrant we get the principal angle by taking (180º - related acute angle) • In the third quadrant we can get the principal angle by taking (180º + related acute angle) • In the fourth quadrant we can get the principal angle by taking (360º - related acute angle)

Let’s work with some numbers! Principal angle 60º 1 0.8660 0.5 1.7320 Related acute angle (none) Principal angle 135º 2 0.7071 -0.7071 -1 Related acute angle 45º 0.7071 0.7071 1 Principal angle 220º 3 -0.6427 -0.7760 0.8391 Related acute angle 40º 0.6427 0.7760 0.8391 Principal angle 300º 4 -0.8660 0.5 -1.7320 Related acute angle 60º 0.5 0.8660 1.7320

Quadrant 1 Sinθ is positive Cosθ is positive Tanθ is positive θ

Quadrant 2 Sinθ = sin (180° - θ) -Cosθ = cos (180° - θ) -Tanθ = tan (180° - θ) (180° - θ) θ

Quadrant 3 -Sinθ = sin (180° + θ) -Cosθ = cos (180° + θ) Tanθ = tan (180° + θ) (180° + θ) θ

Quadrant 4 -Sinθ = sin (360° - θ) Cosθ = cos (360° - θ) -Tanθ = tan (360° - θ) (360° - θ) θ

Summary Only sine is positive All ratios are positive S A Only tangent is positive Only cosine is positive T C

Summary • For any principal angle greater than 90 , the values of the primary trig ratios are either the same as, or the negatives of, the ratios for the related acute angle • When solving for angles greater than 90 , the related acute angle is used to find the related trigonometric ratio. The CAST rule is used to determine the sign of the ratio CAST Rule Quadrant I Quadrant II Sine All 1800 - q q 3600 - q 1800 + q Cosine Tangent Quadrant IV Quadrant III

Example1. Point P(-3,4) is on the terminal arm of an angle in standard position. a)Sketch the principal angle θ b) Determine the value of the related acute angle to the nearest degree c) What is the measure of θ to the nearest degree?

P(-3,4) 4 θ β -3 Solution a) Point P(-3,4) is in quadrant 2, so the principal angle θ terminates in quadrant 2. b) The related acute angle β can be used as part of a right triangle with sides of 3 and 4. We can figure out β using SOHCAHTOA. Note…..Whenever we make a triangle such as the one above there is something important to remember… THE HYPOTENUSE will always be expressed as a positive value, regardless of the quadrant in which it occurs!! Lets look at an example….

Example 2 • Point (3,-4) is on the terminal arm of an angle in standard position • What are the values of the primary trigonometric functions? • What is the measure of the principal angle θ to the nearest degree?

θ 3 -4 r =5 Assuming that you can draw a circle around the x-y axis, with your point lying somewhere on the perimeter, then it would follow that the hypotenuse of our right angled triangle would be the same as the radius of the circle. Solution Using pythagorean theorem, we find that r = 5 (note it is positive regardless of the quadrant. Using these values, then To evaluate B, select cosine and solve for B. Using cos B gives us ……….

From the sketch, clearly θ is not 53°. This angle is the related acute angle. In this case θ = 360°-53° = 307° Just as a side note….once again notice that if you take the cos of 307° you get 0.6018 and if you take the cos of 53° you also get 0.6018

Ok, hmk is pg. 299 #1-6, 8,-10 Wait for it, wait for it… Well Ross, what is it? WOOOOW!

5.3 and 5.4 Evaluating Trig Ratios for Angles between 0 and 360