Activity 4-2: Trig Ratios of Any Angles Part 1: Review

Activity 4-2: Trig Ratios of Any Angles Part 1: Review of Understanding Angles In grade 11 you learned how to find the trigonometric ratios of any angle Before we can do this we must first define some key features of angles x y Terminal Arm θ Initial Arm Initial Arm: the ray that defines the beginning of the angle. Standard Position: when the initial arm lies on the positive x-axis and the vertex of the angle is at the origin (0,0). Terminal Arm: the ray that defines the end of the angle

Activity 4-2: Trig Ratios of Any Angles Part 1: Review of Understanding Angles Angles can either be positive or negative If the terminal arm rotates Counter clockwise=POSTIVE, Clockwise=NEGATIVE NEGATIVE ANGLE θ POSITIVE ANGLE θ π/2 rad 90o Terminal Arm π/4 rad π rad 180o θ 0 rad 0o Terminal Arm Initial Arm θ -3π/4 rad 3π/2 rad 270o

Activity 4-2: Trig Ratios of Any Angles Part 1: Review of Understanding Angles To understand angles we also need to know the terms: Principal Angle and Acute Angle x y θ=7π/4 Principal Angle: the angle between 0° and 360° Related Acute Angle: the angle formed between the terminal arm and the x-axis, and has a measure of between 0° and 90° θ=π/4

Activity 4-2: Trig Ratios of Any Angles Part 1: Review of Understanding Angles Finally, let us review the trigonometric ratios:

Activity 4-2: Trig Ratios of Any Angles Part 1: Review of Understanding Angles Let us use trigonometry to calculate angles in standard position Find the value of angle θ in radians x y Since you have x and y you must use the tangent ratio: tan β = y/x Solve for β: tan β=3/4 β = tan-1(3/4) β = 0.644 rads Label the triangle using positive values for x and y: x=4 and y=3 and label the hypotenuse as r Find the principal angle θ: θ = 2π – 0.644 θ = 5.64 rads θ Create an acute right triangle at x=3 4 β 3 r (4, -3)

Activity 4-2: Trig Ratios of Any Angles Part 1: Review of Understanding Angles Try this example: Find the primary trig ratios and the value of angle θ in radians ANGLE sin β = y/r sin β = 2/(2√5)=1/√5 β = sin-1(1/√5) β = 0.464 rad OR 26.57o .: θ = π – 0.464 = 2.678rad OR = 1800 – 26.57o 153.4o RATIO sinθ= 2/(2√5)=1/ √5 r2 = x2 + y2 r2 = (-4)2 + (2)2 r2 = (16) + (4) r2 = 20 r = 2√5 r ≈ 4.47 ANGLE cos β = x/r Use positive values for x, y, and r when finding the acute angle cos β = 4/(2√5)=2/√5 β = 0.464 rad OR 26.57o .: θ = π – 0.464 = 2.678rad OR = 180o – 26.57o =153.4o RATIO cosθ= -4/(2√5)=-2/√5 To find the trig ratios find the value of r ANGLE tan β = y/x Use positive values for x, y, and r when finding the acute angle tan β = 2/(4)=1/2 β = 0.464 rad OR 26.57o .: θ = π – 0.464 = 2.678rad OR = 180o – 26.57o =153.4o RATIO tanθ= 2/(-4)=-1/2 x y (-4, 2) θ r=2√5 r 2 β -4

Activity 4-2: Trig Ratios of Any Angles Part 1: Review of Understanding Angles To summarize: x y In relation to your diagram, if the angle IS in the FOURTH QUADRANT: Your angle is (2π – acute angle) and the COSINE ratio is only POSITIVE ratio. In relation to your diagram, if the angle IS in the SECOND QUADRANT: Your angle is (π – acute angle) and the SINE ratio is only POSITIVE ratio. ALWAYS draw your angle using the terminal and initial arm When finding the acute angle use the positive values for x and y In relation to your diagram, if the angle IS in the THIRD QUADRANT: Your angle is (π + acute angle) and the TANGENT ratio is only POSITIVE ratio. In relation to your diagram, if the angle IS in the FIRST QUADRANT: The acute angle is your angle and ALL the trig ratios are POSITIVE QUADRANT 1: ALL RATIOS ARE POSITIVE QUADRANT 2: SINE RATIO IS POSITIVE θ θ θ θ QUADRANT 4: COSINE RATIO IS POSITIVE QUADRANT 3: TANGENT RATIO IS POSITIVE

Activity 4-2: Trig Ratios of Any Angles Part 1: Review of Understanding Angles You have completed the first section of today’s activity. Go back to the activity page and complete the questions assigned in this section.