2.2 Trig Ratios of Any Angle (x, y, r) Chapter 1 Trigonometry 2.2 Trig Ratios of Any Angle (x, y, r) 30° - 60° - 90° Triangle 45° - 45° - 90° Triangle 300 450 2 1 600 450 1 1 Pre-Calculus 11

Finding the Trig Ratios of an Angle in Standard Position Suppose angle θ is an angle in standard position. Choose a point (x, y) on the terminal arm, at a distance r from the origin. P(x, y) r y θ x As x and y increase r also increases. If r stays the same x and y are inversely proportional. r2 = x2 + y2 r = √x2 + y2 x, y, and r represent the sides of a right triangle. For angle θ, x = adjacent, y = opposite and r = hypotenuse Pre-Calculus 11

Finding the Trig Ratios of an Angle in Standard Position Suppose angle θ is an angle in standard position. How are the ratios affected if we choose the point (-x, y) on the terminal arm? P(-x, y) r y θ 𝜽R -x r2 = (-x)2 + y2 The horizontal and vertical lengths are considered as directed distances (so they can be positive or negative depending on which quadrant the terminal arm lies in) Pre-Calculus 11

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Sine All Tangent Cosine ( -, + ) ( +, + ) ( -, - ) ( +, - ) How can you tell if the ratios will be positive or negative? Recall the CAST Rule Sine All ( -, + ) ( +, + ) Tangent Cosine ( -, - ) ( +, - ) Pre-Calculus 11

Finding Trigonometric Ratios of Angles in Standard Position Sine is positive. All are positive. r is always positive Tan is positive. Cos is positive. Pre-Calculus 11

Determine the sign of the ratio. 1. sin 1270 2. tan 240 The Cast Rule Determine the sign of the ratio. 1. sin 1270     2. tan 240 3. cos 2600 4. tan 1450 5. cos  970     6. cos 450 7. sin 3140 8. cos 3150 Positive Positive Negative Negative Negative Positive Negative Positive Pre-Calculus 11

The Cast Rule I II I IV I III II IV i ) Sine ratios have positive values in quadrants _____ and _____ . ii) Cosine ratios have positive values in quadrants _____ and _____ . iii) Tangent ratios have positive values in quadrants _____ and _____ . iv) sinθ > 0 and cos θ < 0 ______ v) tan θ < 0 and sin θ < 0 ______ I IV I III II IV Pre-Calculus 11

Finding the Exact Trig Ratios of an Angle in Standard Position Given point P(5, 12) on the terminal arm calculate the exact values of the primary trig ratios. P(5, 12) 13 12 θ 5 Pre-Calculus 11

Finding the Trig Ratios of an Angle in Standard Position The point P(-2, 3) is on the terminal arm of θ .in standard position. Determine the exact value and approximate value of the trigonometric ratios for angle θ. Exact Values P(-2, 3) 3 θ Approximations 𝜽R -2 r2 = x2 + y2 r2 = (-2)2 + (3)2 r2 = 4 + 9 r2 = 13 r = √ 13 ← since we did not have to round the decimal here it is technically an exact value. However, we would usually leave as a radical (a/b) exact values are fractions: ⅓ is an exact value. 0.333 is not Exact values are more accurate than approximations. However, they may not be as practical. Ex asking someone to cut a piece of paper at cm may be difficult for that piece to visualize that measurement Pre-Calculus 11

x -5 7 Determine the exact value of the trig ratios given The terminal arm must be in Quadrant III, since sin𝜽 is negative and tan𝜽 is positive x θ -5 7 Use pythagorean theorem to find x (note x < 0 in QIII) Pre-Calculus 11

1. sin 250 = 2. cos 1210 = 3. tan 3350 = 4. sin 00 = 5. tan 900 = Calculator: Determine Approximate Trig Ratios (four decimal places) 1. sin 250 = 2. cos 1210 = 3. tan 3350 = 4. sin 00 = 5. tan 900 = Why is tan 90° undefined? Pre-Calculus 11

2.2 B Trig Ratios of Any Angle Quadrantal Angles and Solve for the Angle The reference angles for angles in standard position 150° and 210° are equal. Does this imply that ? ? ? ref 30° ref 30° Q II, sine is +ve Q III, sine is -ve How does sin 150° compare to sin 30°? Pre-Calculus 11

Quadrantal Angle If an angle is standard position with the terminal arm on the x or y axis we call the angle a quadrantal angle. Examples include angles at 0o, 900, 180o, 270o, 360o, 450o, … etc. You should be able to find the trig functions of quadrantal angles and angles with a reference angle of 30°, 45°, and 60° without using a calculator. Pre-Calculus 11

Values of Trigonometric Functions for Quadrantal Angles

Quadrantal Angles Example P(0, 3) Q(-4, 0) r2 = x2 + y2 90° 180° 0° , 360° Q(-4, 0) r2 = x2 + y2 270° Pre-Calculus 11

Determine the Measure of an Angle Given a Trig Ratio Solve for angle θ given 00 ≤ θ < 3600 Sin𝜽 is positive in both quadrant I and II so we will have 2 possible answers 2 2 1 1 θ 𝜽R θ We know from the previous chapter that sin 300 = 1/2 , so the reference angle must be 300 Sohcahtoa QI QII θR = 300 Reference Angle θR = 300 Reference Angle Angle in Standard Position Angle in Standard Position θ = 300 θ = 1500 Pre-Calculus 11

Determine the Measure of an Angle Given a Trig Ratio 00 ≤ θ < 3600 nearest degree QI or QII Solve for angle θ given 5 5 3 3 θ 𝜽R θ When the trig ratio cannot be related to the special triangles or quadrantal angle we need to use the Inverse Trig Function on our calculator to determine the angle. θR = 370 Reference Angle θR = 370 Reference Angle Angle in Standard Position Angle in Standard Position θ = 370 θ = 1430 Pre-Calculus 11

Inverse Trig Functions The inverse trigonometric functions sin-1(x), cos-1(x), and tan-1(x), are used to find the unknown measure of an angle of a right triangle when two side lengths (ratio) are known Note the “-1” in this function is not an exponent. It is simply a notation that we use to tell us we are dealing with an inverse trig function. If we wanted a function to denote 1 over cosine we would use: We can also use the notation arcsin(x), arccos(x), and arctan(x) to denote an inverse trig function

Inverse Trig Functions To evaluate inverse trig remember that the following statements are equivalent: So, when we evaluate an inverse trig function we are asking “what angle, 𝜽, do we plug into the original trig function to get x ? “ Example In other words, what angle, 𝜽, do we need to plug into cosine to get ? We can use our knowledge of special triangle or a calculator to solve this problem When using the inverse function on your calcular the output value (answer) is treated as the reference angle. This is because there is 2 possible answers, the calculator can only give us 1. Cosine is positive in quadrant I and IV so, and

Inverse Trig Functions Example Given , determine the measure of x, to the nearest tenth of a degree When evaluating a trig function in which the ratio is negative we will ignore the negative sign when determining the reference angle. The negative value will be accounted for when we determine the angle in standard position. ← use only positive numbers in this formula Note: ≈ means “approximately” Cosine is negative in quadrants II and III In Quadrant II: In Quadrant III: Therefor, x = 169.9° and x = 190.1°

Determine the Measure of the Angle Given the Exact Ratio Solve for each angle θ given a specific trig ratio. 00 ≤ θ < 3600 QI QII QI QIII θ = 450, θ = 300 , 2100 1350 RA = 300 RA = 450 QI QII QI QIV θ = 300, 1500 θ = 600 , 3000 RA = 600 RA = 300 QII QIII QII QIV θ = 1350 , 2250 θ = 1200 , 3000 RA = 450 RA = 600 QII QIII QII QIII θ = 1200 , 2400 θ = 1500 , 2100 RA = 600 RA = 300 Pre-Calculus 11

Determine the Measure of the Angle Given the Approximate Ratio Determine the measure of angle A, to the nearest degree: 00 ≤ A < 3600 Enter a positive ratio in your calculator R A Quadrants 340 I 340 II 1460 sinA = 0.5632 410 II 1390 III 2210 cosA = -0.7542 570 II 1230 IV 3030 tanA = -1.5643 cosA = 0.5986 530 I 530 IV 3070 3000 III 2400 IV sinA = -0.8667 600 tanA = 0.5965 310 I 310 III 2110 Pre-Calculus 11

Assignment Suggested Questions Page 96: 1-9, 14-16, 19, 26-29 Pre-Calculus 11