2.1 Angles in Standard Position Chapter 2 Trigonometry 2.1 Angles in Standard Position In geometry, an angle is formed by two rays with a common endpoint. In trigonometry, angles may be interpreted as rotations of a ray. Vertex Initial Arm Terminal Arm Pre-Calculus 11

Angles in Standard Position Yes Examples No Examples Where would these go? No Yes No How would you describe an angle in standard position? Pre-Calculus 11

Angles in Standard Position Chapter Identify the angles sketched in standard position. Check answer Pre-Calculus 11

Angles in Standard Position An angle θ is said to be in standard position if its vertex is at the origin of a rectangular coordinate system and its initial arm coincides with the positive x-axis. The rotation of angle θ is measured in degrees. Initial arm Vertex Terminal arm x y Pre-Calculus 11

Example Sketch each angle in standard position. State the quadrant in which the terminal arm lies. 170° 50° 200° 300° QII QI QIV QIII Pre-Calculus 11

Reference Angles, 𝜃R An acute angle formed between the terminal arm and x-axis. This angle is always positive and measures between 0o and 90o Pre-Calculus 11

Determining Reference Angles by Quadrant

Reflections of Reference Angles Determine the Angle in Standard Position w/ a Reference angle of 30° 300 300 1500 300 Angle in Standard Position_______ Angle in Standard Position_______ 300 300 Angle in Standard Position_______ 2100 3300 Angle in Standard Position_______ Pre-Calculus 11

Reflections of Reference Angles Determine the Angle in Standard Position w/ a Reference angle of 60° 600 600 600 1200 Angle in Standard Position _______ Angle in Standard Position _______ 600 600 3000 Angle in Standard Position _______ 2400 Angle in Standard Position _______ Pre-Calculus 11

Reflections of Reference Angles Determine the Angle in Standard Position w/ a Reference angle of 45° 45° 45° 135° 45° Angle in Standard Position _______ Angle in Standard Position _______ 45° 45° 315° 225° Angle in Standard Position _______ Angle in Standard Position _______ Pre-Calculus 11

Sketching Angles in Standard Position Sketch the following angles in standard position. State the size of the reference angle. In which quadrant does the terminal arm lie? a) 1500 b) 2900 c) 2150 Quadrant II Quadrant IV III Quadrant Reference Angle 300 Reference Angle 700 Reference Angle 350 Pre-Calculus 11

Drawing Angles in Standard Position Given a Point on the Terminal Arm a) Draw an angle, θ, in standard position such that the point P(-4, 3) lies on the terminal arm of an angle θ P(-4, 3) 3 θR θ -4 b) Suppose the point P(-4, 3) was reflected about the y - axis P(-4, 3) θ Q(4, 3) θR θR Pre-Calculus 11

Drawing Angles in Standard Position Given a Point on the Terminal Arm c) Suppose the point P(-4, 3) was reflected about the x-axis P(-4, 3) θ θR θR R(-4, -3) d) Suppose the point P(-4, 3) was reflected about the x-axis and y-axis P(-4, 3) θ θR θR S(4, -3) Pre-Calculus 11

Torso Angle - Fast Torso Angle - Touring Torso angle is very dependent upon the cyclists choice of performance and comfort. A lower position is more aerodynamic as frontal surface area is reduced. 30° to 40° is a good compromise of performance and comfort but does rely on reasonably good flexibility to lower back and hamstrings. Torso Angle - Touring A more relaxed torso angle will take the pressure off the lower back, hamstrings and the neck and distribute loads from hands to seat. 40° to 50° is a suitable angle for longer distances where comfort is the priority over speed. Pre-Calculus 11

Reference Angles Determine the measure of the reference angle. Angle in Standard Position (θ) Quadrant Reference Angle (θR) 165° 320° 250° 60° II 15° IV 40° III 70° I 60° Reference Angle (θR) Quadrant Angle in Standard Position (θ) 85° III 46° I 37° IV 52° II Determine the measure of the angle in standard position. 265° 46° 323° 128° Pre-Calculus 11

Chapter 1 Sequences and Series 2.1 B Angles in Standard Position - Exact Values Pre-Calculus 11

The Primary Trigonometric Ratios Trigonometry compares the ratios of the sides in a right triangle. The Primary Trigonometric Ratios There are three primary trig ratios: sine cosine tangent Opposite the right-angle hypotenuse Opposite the angle. opposite adjacent Next to the angle 30° 1 2 Pre-Calculus 11

Trig Equations sin 30º= trig function angle trig ratio Knowing the measure of the reference angle, can you label the triangle? 300 2 1 Pre-Calculus 11

Recall the Special Right Triangles and Pythagorean Theorem Pre-Calculus 11

Exact Values for Trig Ratios of Special Right Angles Example For the following triangle to determine the three trig ratios for 45° Use what you know about the ratio of sides in a 45-45-90 triangle or pythagorean theorem to determine the value of c. Example For the following triangle to determine the three trig ratios for 30° and 60°

Example Exact Value - Approximation - Exact Values of Trig Ratios Exact Value Vs Decimal Approximation Example Exact Value - Approximation - Pre-Calculus 11

Comparing Trigonometric Ratios using Common angles Quadrant Sin Cos Tan 30° 150° 210° 330° Note: I II III IV What do the angles have in common? The reference angle is θR = 30° What do you notice about the ratios of the lengths of sides? They all have the same reference angle For each trig function the ratio is the same only the sign changes Q I: all +ve, QII sin +ve, QIII tan +ve, QIV cos +ve The side length have the same ratios as a 30-60-90 triangle, Make a conjecture to determine the sign of the trig ratio for each quadrant QI: all trig ratios are positive QII: only sin is positive, QIII: only tan is positive QIV: only cos is positive Pre-Calculus 11

Comparing Trigonometric Ratios using Common angles Quadrant Sin Cos Tan 60° 120° 240° 300° θR = 60° I II III IV Angle Quadrant Sin Cos Tan 45° 135° 225° 315° θR = 45° I Note: II III IV How can we use the quadrant and trig function to determine the sign of the trig ratio? Pre-Calculus 11

The Cast Rule The CAST diagram helps us to see which quadrants the trig ratios are positive

McGraw-Hill Ryerson Precalculus 11 Page 82 Example 4 Calculate the horizontal distance to the midline, labeled a. a Which trig ratio would you use to determine the length of side a? The exact horizontal distance is 10 10 cm. 60° a Pre-Calculus 11

Using Exact Values Homework 2. cos 450 = 3. tan 450 = 4. sin 600 = 5. sin 1500 = 6. cos 1200 = RA = 300 RA = 600 7. tan 1350 = RA = reference angle 8. tan 1200 = RA = 450 RA = 600 9. sin 1350 = 10. cos 1500 = RA = 450 RA = 300 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 the side lengths 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 input 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°

Assignment Suggested Questions Page 83: 1-7, 9, 10, 13, 14, 16, 18, 23, 24 Pre-Calculus 11