SECTION 2 2 TRIG RATIOS WITH ANY ANGLE

SECTION 2. 2 TRIG RATIOS WITH ANY ANGLE © Copyright all rights reserved to Homework depot: www. BCMath. ca

I) REVIEW: SOH-CAH-TOA 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

II) SPECIAL TRIANGLES There are two types of special triangles: � 30°, 60°, 90° triangle (Equilateral Triangle) and � 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 height of the triangle Use Pythagorean Thm. to find the hypotenuse 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

III) TRIG RATIO FOR ANY ANGLE When given the angle in std. position, we can find the coordinates of any point on the circumference using trigonometry 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 th nd 4 Quadrant Only Cosine ofall inin Tangent ofangles 3 Quadrant willbe bepositive sine ofofangles in the 2 the will be positive sin/cos/tan angles inthe 1 rdst. Quadrant rdth. Quad nd Quad Sine & Tangent the 4 willbe benegative Cosine ininthe will Cosine & Tangent in 3 the 2 Quad will be negative © 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 S A S A T C T C © Copyright all rights reserved to Homework depot: www. BCMath. ca

Ex: Solve for the angle, given the following trig functions. S A The angle has to be in Q 1 and Q 2 C Inverse sine the equation to find where the angle is The ratio is positive T Find the 2 nd angle in Q 2 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

Practice: Solve for the angle: S A The angle has to be in Q 2 and Q 3 C Inverse cosine the equation to find where the angle is The ratio is negative T Find the 2 nd angle in Q 3 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 Draw the 30°, 60°, 90° triangle Draw the 45°, 90° triangle © Copyright all rights reserved to Homework depot: www. BCMath. ca

VI) FINDING THE COORDINATES OF POINTP 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 The ratio is positive S A T C 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 The ratio is negative S A T C 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 The ratio is negative S A T C 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 The ratio is positive S A T C 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