Activity 4 2 Trig Ratios of Any Angles

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 y l. A rm a in m r Te θ Initial Arm x Initial Arm: the ray that defines Standard Position: when the beginning the angle. initial arm. Arm: lies of on the positive x Terminal the ray that -axis and the vertex of angle the defines the end of the angle is at the origin (0, 0).

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 ANGLEθθ POSITIVE ANGLE π/2 rad 90 o rm rm l. A Te rm in al A π rad 180 o rm e T a in θ π/4 rad Initial Arm θ 3π/2 rad 270 o -3π/4 rad 0 o

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 y θ=7π/4 x θ=π/4 Related Acute Angle: the angle Principal Angle: the terminal angle formed between 0° and 360°and has a arm and the x-axis, measure of between 0° and 90°

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 y θ 4 β r x 3 (4, -3) Find the Label the triangle Since you have x and y Solve forprincipal β: angle θ: use the using positive values you must tan β=3/4 Create an acute right for x 2π and y: x=4 and tangent ratio: θ = – 0. 644 -1 β = tan (3/4) triangle at x=3 the y=3 and label tan β = y/x θ = 5. 64 rads β = 0. 644 rads hypotenuse as r

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 y (-4, 2) 2 r=2√ 5 r -4 β θ x 2 =find ANGLE r. To x 2 +the y 2 trig ratios 2 =β value sin tan β ===2 y/r y/x cos rfind (-4) βthe x/r + (2)2 of r sin = 2/(2√ 5)=1/√ 5 Use r 2 = β(16) positive + (4) values for -1(1/√ 5) x, and r when finding β r 2 =y, sin 20 the acute angle -1(1/√ 5) β sin r == 2√ 5 ββ == 2/(4)=1/2 cos 4/(2√ 5)=2/√ 5 β 0. 464 rad OR 26. 57 ooo rtan ≈= 4. 47 β = 0. 464 rad OR 26. 57. : θ θ= =π π– – 0. 464 = =. : 2. 678 rad OR OR 0 o o = 180 oo – = – 26. 57 oo 153. 4 =153. 4 oo RATIO tanθ= 2/(2√ 5)=1/ 2/(-4)=-1/2 sinθ= √ 5 cosθ= -4/(2√ 5)=-2/√ 5

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