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Radian measure Trig functions of general angles Finding the reference angle Basic trig identities Calculating trig functions Quiz Review Every real number is the radian measure of some angle Every real number, viewed as a number of radians, represents an angle. The initial side of the angle is a horizontal arrow pointing right from the origin. Rotate that arrow counterclockwise (or clockwise) with each full turn counting as 2π (or −2π) radians. The final position of the arrow is the terminal side of the angle. θ initial side terminal side In the example at the right, the initial side is rotated through 43 of a complete circle. The rotation is suggested by the red dashed arc. The angle θ corresponds to 34 of a complete trip around the circle, and so ◦ θ = 34 · 2π = 3π 2 rad = 270 . y More examples are on the next slide. Stanley Ocken M19500 Precalculus Chapter 6.3 Trigonometric functions of angles x Radian measure Trig functions of general angles Radians Degrees y Finding the reference angle 2π 360 y π 180 x 3π rad 4 = 135◦ π/3 60 y π/4 45 x y −πrad = −180◦ Stanley Ocken π/6 30 x 5π rad 4 = 330◦ x = 225◦ y x − π6 rad = −30◦ Quiz Review y y x Calculating trig functions x 11π rad 6 πrad = 180◦ y − 5π rad = −225◦ 4 π/2 90 Basic trig identities x - 3π rad = −135◦ 4 M19500 Precalculus Chapter 6.3 Trigonometric functions of angles Radian measure Trig functions of general angles Finding the reference angle Basic trig identities Calculating trig functions Quiz Review Trig functions of angles in Quadrant 1 Let θ be any angle. Place its initial side on the positive x-axis. Suppose the arrow tip of the terminal side of θ ispat point P (x, y) in the x,y-plane. Let r = x2 + y 2 be the distance from P (x, y) to the origin. r is positive! y 5 4 P (x, y) (4, 3) 3 The trig functions of angle θ x r y sin(θ) = r tan(θ) = xy cos(θ) = r=5 2 1 r cos θ = x 1 r csc(θ) = sin θ = y 1 x cot(θ) = tan θ = y sec(θ) = 3 1 θ 0 -5 -4 -3 -2 -1 0 1 4 2 3 4 5 -1 In the picture at the right, the arrow tip of the terminal side is atp P (4, 3). Thus x = 4, y = 3, r = x2 + y 2 = 5 and so x 4 r = 5 sin(θ) = yr = 35 tan(θ) = xy = 43 cos(θ) = -2 -3 -4 r 5 x = 4 csc(θ) = yr = 53 cot(θ) = yr = 43 sec(θ) = -5 Stanley Ocken M19500 Precalculus Chapter 6.3 Trigonometric functions of angles x Radian measure Trig functions of general angles Finding the reference angle Basic trig identities Calculating trig functions Quiz Review Trig functions of angles in Quadrants 2,3,4 On this slide we are concerned not with the values of trig functions, but with their signs. Remember that r is always positive, while x and y can be positive or negative. 5 y • cos(θ) = x/r has the same sign as x, which is positive in Q1 and Q4, negative in Q2 and Q3. • sin(θ) = y/r has the same sign as y, which is positive in Q1 and Q2, negative in Q3 and Q4. • tan(θ) = y/x is positive in Q1 and Q3, negative in Q2 and Q4. 4 3 Q2: x Neg; y Pos Q1: x Pos; y Pos 2 1 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 -1 -2 Q3: x Neg; y Neg Q4: x Pos; y Neg -3 -4 -5 Stanley Ocken M19500 Precalculus Chapter 6.3 Trigonometric functions of angles x Radian measure Trig functions of general angles Finding the reference angle Basic trig identities Calculating trig functions Quiz Review Trig functions of angles in Quadrants 2,3,4 On this slide we are concerned not with the values of trig functions, but with their signs. Remember that r is always positive, while x and y can be positive or negative. 5 y • cos(θ) = x/r has the same sign as x, which is positive in Q1 and Q4, negative in Q2 and Q3. • sin(θ) = y/r has the same sign as y, which is positive in Q1 and Q2, negative in Q3 and Q4. • tan(θ) = y/x is positive in Q1 and Q3, negative in Q2 and Q4. You may choose to derive these facts from ASTC, but it’s better to think of the picture. The quadrant by quadrant results are: Quadrant 1: All: cos, sin, tan are +. Quadrant 1 4 P (x = 4, y = 3) 3 2 1 θ 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 -1 -2 -3 -4 -5 Stanley Ocken M19500 Precalculus Chapter 6.3 Trigonometric functions of angles x Radian measure Trig functions of general angles Finding the reference angle Basic trig identities Calculating trig functions Quiz Review Trig functions of angles in Quadrants 2,3,4 On this slide we are concerned not with the values of trig functions, but with their signs. Remember that r is always positive, while x and y can be positive or negative. 5 y • cos(θ) = x/r has the same sign as x, which is positive in Q1 and Q4, negative in Q2 and Q3. • sin(θ) = y/r has the same sign as y, which is positive in Q1 and Q2, negative in Q3 and Q4. • tan(θ) = y/x is positive in Q1 and Q3, negative in Q2 and Q4. You may choose to derive these facts from ASTC, but it’s better to think of the picture. The quadrant by quadrant results are: Quadrant 1: All: cos, sin, tan are +. Quadrant 2: Sin is +, cos and tan are −. Quadrant 2 4 P2 (x = −4, y = 3) 3 θ2 2 1 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 -1 -2 -3 -4 -5 Stanley Ocken M19500 Precalculus Chapter 6.3 Trigonometric functions of angles x Radian measure Trig functions of general angles Finding the reference angle Basic trig identities Calculating trig functions Quiz Review Trig functions of angles in Quadrants 2,3,4 On this slide we are concerned not with the values of trig functions, but with their signs. Remember that r is always positive, while x and y can be positive or negative. 5 y • cos(θ) = x/r has the same sign as x, which is positive in Q1 and Q4, negative in Q2 and Q3. • sin(θ) = y/r has the same sign as y, which is positive in Q1 and Q2, negative in Q3 and Q4. • tan(θ) = y/x is positive in Q1 and Q3, negative in Q2 and Q4. You may choose to derive these facts from ASTC, but it’s better to think of the picture. The quadrant by quadrant results are: Quadrant 1: All: cos, sin, tan are +. Quadrant 2: Sin is +, cos and tan are −. Quadrant 3: Tan is +, cos and sin are −. 4 3 θ3 2 1 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 -1 -2 -3 P3 (x = −4, y = −3) -4 Quadrant 3 -5 Stanley Ocken M19500 Precalculus Chapter 6.3 Trigonometric functions of angles x Radian measure Trig functions of general angles Finding the reference angle Basic trig identities Calculating trig functions Quiz Review Trig functions of angles in Quadrants 2,3,4 On this slide we are concerned not with the values of trig functions, but with their signs. Remember that r is always positive, while x and y can be positive or negative. 5 y • cos(θ) = x/r has the same sign as x, which is positive in Q1 and Q4, negative in Q2 and Q3. • sin(θ) = y/r has the same sign as y, which is positive in Q1 and Q2, negative in Q3 and Q4. • tan(θ) = y/x is positive in Q1 and Q3, negative in Q2 and Q4. You may choose to derive these facts from ASTC, but it’s better to think of the picture. The quadrant by quadrant results are: Quadrant 1: All: cos, sin, tan are +. Quadrant 2: Sin is +, cos and tan are −. Quadrant 3: Tan is +, cos and sin are −. Quadrant 4: Cos is +, sin and tan are −. 4 3 θ4 2 1 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 -1 -2 -3 P4 (x = 4, y = −3) -4 Quadrant 4 -5 Stanley Ocken x M19500 Precalculus Chapter 6.3 Trigonometric functions of angles Radian measure Trig functions of general angles Finding the reference angle Basic trig identities Calculating trig functions y 5 We are now going to calculate trig functions of closely related angles. Start with angle θ from before. It will reappear in the next 3 pictures. Quiz Review 4 P(4,3) 3 cos(θ) = xr = 45 sin(θ) = yr = 53 tan(θ) = xy = 43 2 1 r= 0 In this example, θ is a Quadrant 1 angle. This means: θ is an angle whose terminal side lies in Quadrant 1. All trig functions of θ are positive. θ will reappear in the next 3 slides as the reference angle of angles in Quadrants 2,3, and 4. -5 -4 -3 -2 -1 0 1 5 θ 2 3 4 5 x -1 -2 -3 -4 -5 Definition The reference angle of a given angle is the acute ( < 90◦ ) angle between the x-axis and the given angle’s terminal line. Stanley Ocken The reference angle of angle θ is colored yellow on these slides. It is a positive angle, between 0 and 90◦ . All of its trig functions are positive. M19500 Precalculus Chapter 6.3 Trigonometric functions of angles Radian measure Trig functions of general angles Finding the reference angle Basic trig identities Calculating trig functions Quiz Review ◦ The Quadrant 2 angle θ2 shown below has reference angle θ = 180 p − θ2 The endpoint of the terminal side of θ2 is P2 (−4, 3) Thus x = −4, y = 3, r = (−4)2 + 32 = 5. 4 5 y cos(θ2 ) = xr = −4 5 = −5 y 3 sin(θ2 ) = r = 5 4 3 P2 (−4, 3) tan(θ2 ) = xy = −4 = − 43 3 Since θ2 is a Quadrant 2 (Q2) angle, cos(θ2 ) is negative since x is positive in Q2. sin(θ2 ) is positive since y is positive in Q2. tan(θ2 ) is negative since y/x is negative in Q2. It’s easy to compute the trig functions of θ2 if you already know the trig functions of its reference angle θ. For θ2 in Quadrant 2: cos(θ2 ) = − cos(θ). sin(θ2 ) = + sin(θ). tan(θ2 ) = − tan(θ). Stanley Ocken θ2 2 r= θ -5 -4 -3 -2 5 1 0 -1 0 1 2 3 4 -1 -2 -3 -4 -5 M19500 Precalculus Chapter 6.3 Trigonometric functions of angles 5 x Radian measure Trig functions of general angles Finding the reference angle Basic trig identities Calculating trig functions Quiz Review ◦ The Quadrant 3 angle θ3 shown below has reference angle θ3 − 180 p . The endpoint of the terminal side of θ3 is P3 (−4, −3) Thus x = −4, y = −3, r = (−4)2 + (−3)2 = 5. 4 cos(θ3 ) = xr = −4 5 = −5 y 3 sin(θ3 ) = r = −3 5 = −5 y −3 3 tan(θ3 ) = x = −4 = 4 y 5 4 3 Since θ3 is a Quadrant 3 (Q3) angle, cos(θ3 ) is negative since x is positive in Q3. sin(θ3 ) is negative since y is negative in Q3. tan(θ3 ) is positive since y/x is positive in Q3. It’s easy to compute the trig functions of θ3 if you already know the trig functions of its reference angle θ. For θ3 in Quadrant 3: cos(θ3 ) = − cos(θ) sin(θ3 ) = − sin(θ); tan(θ3 ) = + tan(θ). Stanley Ocken θ3 2 1 -5 -4 -3 -2 -1 0 1 2 3 4 0 θ r= 5-1 -2 -3 P3 (−4, −3) -4 -5 M19500 Precalculus Chapter 6.3 Trigonometric functions of angles 5 x Radian measure Trig functions of general angles Finding the reference angle Basic trig identities Calculating trig functions Quiz Review ◦ The Quadrant 4 angle θ4 shown below has reference angle θ = 360 p − θ4 . The endpoint of the terminal side of θ4 is P4 (4, −3) Thus x = 4, y = −3, r = 42 + (−3)2 = 5. cos(θ4 ) = xr = 45 5 y 3 sin(θ4 ) = yr = −3 = − 5 5 4 3 tan(θ4 ) = xy = −3 4 = −4 3 Since θ4 is a Quadrant 4 (Q4) angle, cos(θ4 ) is positive since x is positive in Q4. sin(θ4 ) is negative since y is negative in Q4. tan(θ4 ) is negative since y/x is negative in Q4. It’s easy to compute the trig functions of θ4 if you already know the trig functions of its reference angle θ. For θ4 in Quadrant 4: cos(θ4 ) = + cos(θ) sin(θ4 ) = − sin(θ); tan(θ4 ) = − tan(θ). Stanley Ocken θ4 2 1 -5 -4 -3 -2 -1 0 1 2 3 4 0 -1 r= θ 5 -2 -3 P4 (4, −3) -4 -5 M19500 Precalculus Chapter 6.3 Trigonometric functions of angles 5 x Radian measure Trig functions of general angles Finding the reference angle Basic trig identities Calculating trig functions Quiz Review Finding the reference angle of a given angle. We abbreviate ”the reference angle of θ2 ” by Ref(θ2 ). It is the angle θ shown in yellow on the previous slides. Ref(θ2 ) is the smaller angle between the terminal line of θ2 and the x-axis. The reference angle is always positive: 0◦ <Ref(θ2 ) < 90◦ Similar statements hold for the angles θ3 and θ4 . Summary: How to find the reference angle of an angle given in degrees • Quadrant 1: If 0◦ < θ1 < 90◦ then Ref(θ1 ) = θ1 . • Quadrant 2: If 90◦ < θ2 < 180◦ then Ref(θ2 ) = 180◦ − θ2 . • Quadrant 3: If 180◦ < θ3 < 270◦ then Ref(θ3 ) = θ3 − 180◦ . • Quadrant 4: If 270◦ < θ4 < 360◦ then Ref(θ4 ) = 360◦ − θ4 . • Angles 0, ±90◦ , ±180◦ .... that are a multiple of 90◦ have undefined reference angle. • If θ < 0 or θ > 360◦ , add or subtract a multiple of 360◦ to/from θ to get an angle θ0 with 0◦ ≤ θ0 < 360◦ . Then Ref(θ) = Ref(θ0 ), computed according to the above rules. Stanley Ocken M19500 Precalculus Chapter 6.3 Trigonometric functions of angles Radian measure Trig functions of general angles Finding the reference angle Basic trig identities Calculating trig functions Quiz Review Summary: How to find the reference angle of an angle given in radians • Quadrant 1: If 0 < θ1 < π2 then Ref(θ1 ) = θ1 . • Quadrant 2: If π2 < θ2 < π then Ref(θ2 ) = π − θ2 . • Quadrant 3: If π < θ3 < 3π 2 then Ref(θ3 ) = θ3 − π. • Quadrant 4: If 3π < θ < 2π then Ref(θ4 ) = 2π − θ4 . 4 2 • Angles 0, ± π2 , ±π, ± 3π .... that are a multiple of π2 have undefined reference angle. 2 • If θ < 0 or θ > 2π, add or subtract a multiple of 2π to/from θ to get an angle θ0 with 0 ≤ θ0 < 2π. Then Ref(θ) = Ref(θ0 ), computed according to the above rules. Example 1: Find the cosines of the following angles: 150◦ , 225◦ , 300◦ Solution: • 150◦ : Terminal line in Q2: Ref(150◦ ) = 180◦ − 150◦ = 30◦ . √ Since cosine is negative in Q2, cos(150◦ ) = − cos(30◦ ) = − 23 • 225◦ : Terminal line in Q3: Ref(225◦ ) = 225◦ − 180◦ = 45◦ . Since cosine is negative in Q3, cos(225◦ ) = − cos(45◦ ) = − √12 • 300◦ : Terminal line in Q4: Ref(300◦ ) = 360◦ − 300◦ = 60◦ . Since cosine is positive in Q4, cos(300◦ ) = + cos(60◦ ) = Stanley Ocken 1 2 M19500 Precalculus Chapter 6.3 Trigonometric functions of angles Radian measure Trig functions of general angles Finding the reference angle Basic trig identities Calculating trig functions Quiz Review The following summary need not be memorized if you understand the previous slides. Given any angle θ, let θ be the reference angle of θ. Trig functions of angles in terms of the reference angle • • • • • • cos(θ) = cos(θ) for angles θ ending in Q1 and Q4. cos(θ) = − cos(θ) for angles θ ending in Q2 and Q3. sin(θ) = sin(θ) for angles θ ending in Q1 and Q2. sin(θ) = − sin(θ) for angles θ ending in Q3 and Q4. tan(θ) = tan(θ) for angles θ ending in Q1 and Q3. tan(θ) = − tan(θ) for angles θ ending in Q2 and Q4. Example 2: Find the reference angle of 920◦ . Solution: 920◦ − 360◦ − 360◦ = 200◦ . Since 180◦ < 200◦ < 270◦ , angle 200◦ is in Q3. Its reference angle is 200◦ − 180◦ = 20◦ . Example 3: Find the reference angle of 19π 3 . Solution: Method 1: Convert to degrees: 19( π3 ) = 19(60◦ ) = 1140◦ . Then 1140◦ − 360◦ − 360◦ − 360◦ = 60◦ is in Q1, and so its reference angle is 60◦ . 19π 19π 18π π Method 2: Keep radians. 19π 3 − 2π − 2π − 2π = 3 − 6π = 3 − 3 = 3 is in Q1 and so its reference angle is π3 . Stanley Ocken M19500 Precalculus Chapter 6.3 Trigonometric functions of angles Radian measure Trig functions of general angles Finding the reference angle Basic trig identities Calculating trig functions Quiz Review The Pythagorean trigonometric identities If the terminal side of angle θ goes from theporigin to P (x, y) • The distance from P to the origin is r = x2 + y 2 , and x2 + y 2 = r2 . • cos(θ) = xr • sin(θ) = yr • tan(θ) = xy r r • sec(θ) = x • csc(θ) = y • cot(θ) = xy In what follows, recall that powers of trig functions are abbreviated somewhat strangely: (cos(θ))2 is written cos2 (θ), for instance. x2 Divide x 2 + r + y 2 r y2 r2 r2 = by = 1 and so y 5 4 P(4,3) 3 r=5 2 y=3 1 θ 0 0 1 x=4 2 3 4 to get (cos(θ))2 + (sin(θ))2 = 1. This is usually written as cos2 (θ) + sin2 (θ) = 1 . Similarly, dividing x2 + y 2 = r2 by x2 gives the identity 1 + tan2 (θ) = sec2 (θ) . Similarly, dividing x2 + y 2 = r2 by y 2 gives the identity 1 + cot2 (θ) = csc2 (θ) . These identities should be memorized for success in this course and/or calculus. Stanley Ocken M19500 Precalculus Chapter 6.3 Trigonometric functions of angles 5 x Radian measure Trig functions of general angles Finding the reference angle Basic trig identities Calculating trig functions Quiz Review Each of the Pythagorean identities relates two trig functions. You can express each one in terms of the other. For example, you could solve cos2 (θ) + sin2 (θ) = 1 for cos(θ) as follows: cos2 (θ) = 1 − sin2 (θ) p cos(θ) = ± 1 − (sin2 θ). The choice of plus or minus depends on the quadrant: p First recallpthat (like any other radical) 1 − (sin2 θ) is positive. Therefore cos(θ) = 1 − sin2 (θ) in Quadrants 1 and 4, where cos(θ) is positive, while p cos(θ) = − 1 − sin2 (θ) in Quadrants 2 and 3, where cos(θ) is negative. Example 4: If θ is a Quadrant 3 angle, express tan(θ) in terms of sin(θ). p Solution: From above, we know that cos(θ) = − 1 − sin2 (θ) in Quadrant 3. Therefore tan(θ) = sin(θ) cos(θ) = √sin(θ) 2 − 1−sin (θ) and so in Q3: tan(θ) = − √ sin(θ)2 1−sin (θ) Check your answer: In Quadrant 3, sin(θ) is negative in Q3 while tan(θ) is positive. Therefore both sides of the boxed answer have the same sign. If they would have different signs, you would need to go back and check for an error. Stanley Ocken M19500 Precalculus Chapter 6.3 Trigonometric functions of angles Radian measure Trig functions of general angles Finding the reference angle Basic trig identities Calculating trig functions Quiz Review Evaluating trigonometric functions r= Example 5: If sec(θ) = − 53 and θ is in Quadrant 2, find the other 5 trig functions of θ. Solution: If the terminal line of θ goes from the origin to P (x, y), located distance r from the origin, then sec(θ) = xr = − 35 . Choose any r and x that satisfy this equation, but remember that r is positive. The easiest solution is r = 5 and x = −3. But x2 + y 2 = r2 . Thus (−3)2 + y 2 = 52 9 + y 2 = 25 and so y = ±4. But (x, y) is in quadrant 2, where y is positive, and so y = 4. 5 checks with the given Therefore x = −3, y = 4, r = 5. Note that sec(θ) = xr = −3 information. The requested five trig functions are y 5 3 P (−3, 4) cos(θ) = xr = −3 5 = −5 4 sin(θ) = yr = 45 3 4 tan(θ) = xy = −3 = − 34 θ 2 x −3 3 cot(θ) = y = 4 = − 4 1 csc(θ) = yr = 54 0 x 5 -5 Stanley Ocken -4 -3 -2 -1 0 1 2 3 4 5 M19500 Precalculus Chapter 6.3 Trigonometric functions of angles Radian measure Trig functions of general angles Finding the reference angle Basic trig identities Calculating trig functions Quiz Review How your calculator figures out cosines Example 6: If θ is the radian measure of an angle, it is shown in Calculus III that if −π/2 ≤ θ ≤ π/2, then an approximate formula for cosine is cos(θ) ≈ 1 − θ2 θ4 θ6 + − 2! 4! 6! You know that cos(π/3) = 21 . Test the accuracy of the above formula with θ = π/3 ≈ 1.047. Solution: Compute by hand with 3 decimal place accuracy, or (if you must) use a calculator to check that 1.0472 1.0474 1.0476 + − ≈ 0.500 1− 2 24 720 Stanley Ocken M19500 Precalculus Chapter 6.3 Trigonometric functions of angles Radian measure Trig functions of general angles Finding the reference angle 2 4 Basic trig identities 6 The functions y = cos(θ) and y = 1 − θ2! + θ4! − θ6! Note that the graphs look identical for − π2 ≤ θ ≤ π2 Calculating trig functions Quiz Review are graphed below. 1 0 −π − π2 0 π π 2 y = cos(θ) -1 y =1− Stanley Ocken 2 θ 2! + 4 θ 4! − 6 θ 6! M19500 Precalculus Chapter 6.3 Trigonometric functions of angles Radian measure Trig functions of general angles Finding the reference angle Basic trig identities Calculating trig functions Quiz Review How close to identical, really? The difference of the functions y = cos(θ) and 2 4 6 y = 1 − θ2! + θ4! − θ6! is graphed below. Look at the scale to the left of the graph! When θ = π/3, the difference equals ≈ 0.00003543467. In other words the formula for cos(π/3) = 0.5 gives the correct answer with better than 99.99 per cent accuracy! 0.001 y =1− −π 0 θ2 2! + − π2 θ4 4! − 0 θ6 6! − cos(θ) π 2 π -0.001 Stanley Ocken M19500 Precalculus Chapter 6.3 Trigonometric functions of angles Radian measure Trig functions of general angles Finding the reference angle Basic trig identities Calculating trig functions Quiz Review Quiz Review Example 1: Find the cosines of the following angles: 150◦ , 225◦ , 300◦ Example 2: Find the reference angle of 920◦ . Example 3: Find the reference angle of 19π 3 . Example 4: If θ is a Quadrant 3 angle, express tan(θ) in terms of sin(θ). Example 5: If sec(θ) = − 35 and θ is in Quadrant 2, find the other 5 trig functions of θ. Stanley Ocken M19500 Precalculus Chapter 6.3 Trigonometric functions of angles