Section 3.2 Unit Circles with Sine and Cosine © Copyright all rights reserved to Homework depot: www.BCMath.ca

I) Review: SOH-CAH-TOA Pythagorean Theorem: Circle Equation: Function Ratio Pythagorean Theorem: Circle Equation: Note: The coordinate of any point on the circumference will satisfy the circle equation © Copyright all rights reserved to Homework depot: www.BCMath.ca

Review: Special Triangles There are two types of special triangles: 30°, 60°, 90° triangle (Equilateral Triangle) and 45°, 45°, 90° triangle (Isosceles-Right Triangle) Equilateral Triangle Isosceles-Right Triangle All sides and angles are equal Two sides are equal and one angle Is equal to 90° Cut it in half Use Pythagorean Thm. to find the hypotenuse of the triangle Use Pythagorean Thm. to find the height of the triangle Isoceles  2 equal angles © Copyright all rights reserved to Homework depot: www.BCMath.ca

Special triangles can be used to find the exact value of sine/ cosine/tangent of basic angles like: 30°, 45°, 60°, and 90° Rather than obtaining a decimal representation, we get the exact value as a fraction using special triangles © Copyright all rights reserved to Homework depot: www.BCMath.ca

Ex: Use the special triangles to determine the exact value for each of the following: © Copyright all rights reserved to Homework depot: www.BCMath.ca

II) Unit Circles: In a unit circle, the radius (terminal arm) is 1 unit long The tip of the terminal arm is given by the coordinates (x,y) the terminal arm can be used to make a right triangle, so the sum of both the “x” & “y” coordinates squared will be equal to 1 Any point on the circumference of a unit circle will satisfy the equation! Using the Pythagorean Thm: © Copyright all rights reserved to Homework depot: www.BCMath.ca

III) Sine & Cosine in Unit Circles: When given the angle in std. position, we can find the coordinates of any point on the circumference using trigonometry Note: r = 1 Y-coordinate X-coordinate The coordinates of any point on the circumference of a unit circle can be represented by: © Copyright all rights reserved to Homework depot: www.BCMath.ca

Ex: Given the angle in standard position in a unit circle, find the coordinates of each of the following points P(x,y). © Copyright all rights reserved to Homework depot: www.BCMath.ca

IV) Sine/Cosine/Tangent in Different Quadrants The sine/cosine/tangent of angles in each of the four quadrants will either be positive or negative Use the reference angle to determine whether if the ratio is either +’ve or –’ve S A T C Only Cosine of angles in the 4th Quadrant will be positive Sine & Tangent in the 4th Quad will be negative Only Tangent of angles in the 3rd Quadrant will be positive Sine & Cosine in the 3rd Quad will be negative Only sine of angles in the 2nd Quadrant will be positive Cosine & Tangent in the 2nd Quad will be negative sin/cos/tan of all angles in the 1st Quadrant will be positive © Copyright all rights reserved to Homework depot: www.BCMath.ca

The previous table can be used to determine which quadrant an angle will be in when given the ratio of a trig function Ex: Given each of the following trig. functions and its ratios, determine which quadrants the angle can be in: The ratio is negative The ratio is positive The ratio is negative A S T C A S T C A S T C © Copyright all rights reserved to Homework depot: www.BCMath.ca

S A T C Ex: Solve for the angle, given the following trig functions. The angle has to be in Q1 and Q2 The ratio is positive Inverse sine the equation to find where the angle is T C Find the 2nd angle in Q2 using the reference angle Note: When given a trig equation, there are usually 2 answers Check: © Copyright all rights reserved to Homework depot: www.BCMath.ca

S A C T Practice: Solve for the angle: The angle has to be in Q2 and Q3 The ratio is negative Inverse cosine the equation to find where the angle is C T Find the 2nd angle in Q3 using the reference angle Note: When given a trig equation, there are usually 2 answers Check: © Copyright all rights reserved to Homework depot: www.BCMath.ca

V) Using Special Triangles to Solve Trig Equations For angles with reference angles of 30°, 45°, 60°, we can use special triangles Draw the angle in standard position Find the reference angle Draw the 30°, 60°, 90° triangle © Copyright all rights reserved to Homework depot: www.BCMath.ca

Draw the angle in standard position Find the reference angle Find the reference angle Draw the 30°, 60°, 90° triangle Draw the 45°, 45°, 90° triangle © Copyright all rights reserved to Homework depot: www.BCMath.ca

Draw the angle in standard position Find the reference angle Find the reference angle Draw the 30°, 60°, 90° triangle Draw the 45°, 45°, 90° triangle © Copyright all rights reserved to Homework depot: www.BCMath.ca

VI) Finding the coordinates of point P on the terminal arm When given the ratio of a trig function, you will be asked to find the coordinates of the endpoint on the terminal arm Method #1) Find the value of the central angle Using central angle, we can find the coordinates of point P by using: Note: There are usually two central angles, use reference angles Method #2) Finding the exact value using the Pythagorean Thm. Create a right triangle in the corresponding quadrants Use the ratios of the sides to find sinθ and cosθ Section 3.5 © Copyright all rights reserved to Homework depot: www.BCMath.ca

Ex: Given that cos θ = 3/5 find all the possible coordinates for point P(x,y) in the unit circle Determine which quadrant θ will be in A S T C The ratio is positive Find the angles Find the coordinates using sine & cosine The possible coordinates for P are: © Copyright all rights reserved to Homework depot: www.BCMath.ca

Practice: Given that show the angle in standard position, and the coordinates of P on the unit circle. Determine which quadrant θ will be in A S T C The ratio is negative Find the angles Find the coordinates using sine & cosine The possible coordinates for P are: © Copyright all rights reserved to Homework depot: www.BCMath.ca

Ex: Given that sin θ = and find the exact value of the coordinates of point P(x,y) in the unit circle Determine which quadrant θ will be in A S T C The ratio is negative Use the ratio to draw a right triangle Find Base (Pythagorus) Find the coordinates using sine & cosine Coordinates of P: © Copyright all rights reserved to Homework depot: www.BCMath.ca

Ex: Given that tan θ = 3/5 find all the possible coordinates for point P(x,y) in the unit circle Determine which quadrant θ will be in A S T C The ratio is positive Find the angles Find the coordinates using sine & cosine The possible coordinates for P are: © Copyright all rights reserved to Homework depot: www.BCMath.ca

Homework: Assignment 3.2