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Lesson #90 –Trig. Equations that require factoring A2.A.68 Solve trigonometric equations for all values of the variable from 0° to 360° Trig Function Is Positive in Quadrants sin Is Negative in Quadrants cos tan Solve the following equations: 1) x 2 3 x 2 2) 2x2 x 6 0 3) 3xy y 0 With any trigonometric equation we are really just following the same steps as if we were solving for a normal variable. Therefore, when a trigonometric equation is quadratic or has two different functions in it, we need to factor to solve it. The only new step is taking the inverse trigonometric function to solve for the angle. You can think about what you would do if there were a normal variable in the place of the trig function at any time to make the problem easier. We write Example: Solve for in the interval 0° ≤ ≤ 360°: sin 2 3sin 2 1) Solve the quadratic as if the entire trig. function was a variable. 2) When you are to the point where you have isolated the trig function, use ASTC to determine the quadrants for your answers. (You will have to do this twice since your factoring creates two equations.) 3) Use the inverse trig function to solve for the angles. 4) Put your answers in the quadrants you chose. 1 sin( ) 2 as sin 2 . When you enter it in your calculator, you can simply enter sin . 2 Do not try to enter it with the 2 between sine and ø. 1. Solve: cos 2 x cos x on the interval 0 x 360 2. Find to the nearest degree, all values of in the interval 0 360 that satisfy the equation 6 cos 2 1 cos . 3. If is an angle in 0 360 and tan 2 4 0 , what is the value of to the nearest degree? 2 4. Solve: 3cos x sin x sin x 0 on the interval 0 x 360 . Round to the nearest minute. 5. Challenge Problem: +2 homework points Solve: 𝑠𝑖𝑛 𝑥 − 6𝑠𝑖𝑛𝑥 + 4 = 0 on the interval 0 x 360 . Round to the nearest degree. 2 3 Lesson #91 – Trigonometric Identities A2.A.67 Justify the Pythagorean identities A2.A.58 Know and apply the co-function and reciprocal relationships between trigonometric ratios A Trig Identity is a trig equation that is always true, no matter the value of the variable. Identities that equal each other can be substituted for each other to help solve other trig equations. You already know the trig. identities below. Angles add up to 90° Reciprocal Identities C. sec = D. csc = E. cot = Cofunction Identities Quotient Identities A. tan = B. cot = F. sin = G. cos = H. cos I. tan = J. M. sec = K. cot = L. N. csc = O. Hints for the following problems It usually helps to write everything in terms of sinø and cosø. Use the quotient identity for cotø, not the reciprocal identity. Whenever you see a trig. function with 90-ø, use a cofunction identity. Be CREATIVE and PERSISTENT. You might not get it the first time, so try something else. Simplifying, Solving, and Proving with the Basic Identities 1. 3. Simplify the expression to a single trigonometric function: (cos )(tan ). 2. Solve for x. (Assume the angle measures are degrees) Sin(3x+7)°=Cos(6x+2)° 4. Simplify the expression to a single trigonometric function cot y csc y . Simplify the expression to a single trigonometric function: cos sin(90 ) 4 . 5. Simplify the expression to a single trigonometric function: 7. sec csc . Solve for x. (Assume the angle measures are degrees) 6. 8. 9. Simplify the expression. Leave your answer in terms of sinø and cosø: sec csc 11. 13. Prove that cot csc(90 ) csc Prove that sin (csc 1) tan (cot cos ) If is ø acute, and cos following that are true: tan( x2 4 x) cot(30) cofunctions Prove that tan csc sec 4 circle each of the of the 5 4 5 4 4 sin 90 csc(90 ) 5 5 5 5 csc sec(90 ) 4 4 5 5 sec csc(90 ) 4 4 2 10. If A is acute and tan A , then 3 2 1) cot A 3 1 2) cot A 3 2 3) cot(90 A) 3 1 4) cot(90 A) 3 12. If cos72 sin x , find the number of degrees in the sin 4 5 sec(90 ) measure of acute angle x. 14. The expression (sec2 )(cot 2 )(sin ) is equivalent to 5 5) 6) 7) 8) sin cos csc sec 2 The Pythagorean Identity The picture to the right is the unit circle. Label the point on the circle (x,y). Label the angle ø. Label the base and height of the triangle in terms of x and y. Write the length of the radius. Substitute the sides of the triangle you have labeled into the Pythagorean Theorem. _______________ -4 -3 -2 From your study of the unit circle, you know that x= _______ and y= _________. Substitute these trig functions into your Pythagorean Theorem equation. ____________________. This is the Pythagorean Identity. You can also solve for sin 2 : _______________. Other Pythagorean Identities 2nd Pythagorean Identity (Divide by cos 2 x ) 1 0.5 -1 1 -0.5 -1 -1.5 Basic Pythagorean Identity -2 In the same way you can solve for cos 2 : _______________. Basic Pythagorean Identity 1.5 -2.5 -3 3rd Pythagorean Identity (Divide by sin 2 x ) In this course we will not do too much with the 2nd and 3rd Pythagorean Identities, but you do need to know how to justify them. In other words, you need to know how to get them from the basic Pythagorean Identity. 6 The flowchart below will help you remember the connections between the three Pythagorean Identities. It will also help you remember that the identities in the small textboxes are just different ways of writing the basic, second, and third identities. Basic Pythagorean Identity Second Pythagorean Identity (with tanθ and secθ) sin2θ = cos2θ = 7 Third Pythagorean Identity (with cotθ and cscθ) Lesson #92 – Trigonometric Sum and Difference Identities A2.A.76 Apply the angle sum and difference formulas for trigonometric functions All of the identities from yesterday need to be memorized. Hopefully you already know the reciprocal, quotient, and cofunction identities from earlier this year. For the next two days we will learn identities that will be given to you on the A2&T Reference Sheet. You will want to have your purple sheet handy! a. Sin30°= b. Sin45°= c. Sin60°= Cos30°= e. Cos 45°= f. Cos 60°= g. Tan30°= h. Tan 45°= i. Tan 60°= d. If cos A 5 ,and 270°<A<360°, find sinA. 13 The six trigonometric identities below are given to you on the reference sheet. You do not need to memorize them. We will just be learning how to work with them in this course. Functions of the Sum of Two Angles Functions of the Difference of Two Angles sin( A B) sin A cos B cos A sin B sin( A B) sin A cos B cos A sin B cos( A B) cos A cos B sin A sin B cos( A B) cos A cos B sin A sin B tan( A B) tan A tan B 1 tan A tan B tan( A B) tan A tan B 1 tan A tan B 1. Find and expression for the exact value of sin 75 in simplest radical form. 2. Determine the value of cos15 in simplest radical form. Why does it make sense that sin(75°)=cos(15°)? 8 75= 30 + 45 Therefore you can use sin(45+30) to find the value of sin(75). This does not mean sin75° = sin45°+ sin30°. You MUST use the formula. When you plug in values for these formulas it is important to figure out if you are plugging in an angle for just A or B or if you are plugging in a sine or cosine value for the entire sinA, cosA, SinB, or CosB. For two angles, A and B, it is known that 0 A 90 and 90 B 180 . If cos A 3 7 and sin B 5 25 find the following values in simplest form. 3. sin( A B) 4. cos( A B) 5. Which of the following is equivalent to cos(100)cos(20) sin(100)sin 20 ? 1) cos(120) 2) sin(120) 3) cos(80) 4) sin(80) 6. Which of the following is not equivalent to sin(70) ? 1) 3) 1 cos2 70 sin80 cos10 cos80 sin10 2) sin 40 cos30 cos 40 sin30 4) sin 60 cos10 cos60 sin10 9 7. Find the exact value of tan(75°). 8. If tan A 9. If cos A 1 4 and tan B 5 Find the value of tan(A+B). 3 13 and 270 A 360 and tan B 2.5 Find the value of tan(A-B). 13 10 1. Match each of the trigonometric values in column A with an equivalent expression in column B. Column A Column B 1) sin 35 a) cos50 cos15 sin50 sin15 2) cos35 b) sin50 cos15 cos50 sin15 3) sin 65 c) sin50 cos15 cos50 sin15 4) cos 65 d) cos50 cos15 sin50 sin15 2. Which of the following is equivalent to cos75 cos 25 sin 75 sin 25 ? 1) sin100 2) cos100 3) sin 50 4) cos50 3. Which of the following gives the exact value of cos 75 ? 1) 3) 6 2 4 1 3 4 2) 3 2 2 4) 2 6 2 4. The value of sin80 can be expressed in terms of sin 40 and cos 40 as 5) sin 2 40 cos 2 40 6) 2sin 40 cos 40 7) 1 cos 2 40 8) sin 2 40 cos 2 40 11 5. For two angles, A and B, it is known that 90 A 180 and 270 B 360 . If cos A cos B 4 find each of the following values in exact, simplest form. 5 (a) sin(A+B) (b) sin (A-B) (c) cos(A+B) (d) cos (A-B) 6. For two angles, A and B, it is known that 0 A 90 and 90 B 180 . If sin A sin B 12 and 13 1 , find sin(A+B) in simplest radical form. 2 12 5 and 3 Lesson #93 – Trigonometric Double and Half Angle Identities A2.A.77 Apply the double-angle and half-angle formulas for trigonometric functions Functions of the Double Angle sin 2 A 2sin A cos A 2 tan A tan 2 A 1 tan 2 A cos 2 A cos 2 A sin 2 A c os 2 A 2cos 2 A 1 cos 2 A 1 2sin 2 A Options are a good thing. You can use any of the three cos(2A) identities. I always use the 2 nd or 3rd option depending on if the problem gives me a sine value or a cosine value. Just remember, it does not really matter – JUST PICK ONE! 1. Verify the identity for sin2A using A 30 and 2 A 60 2. For an angle, A, it is known that 90 A 180 . If cos A (a) sin(2A) (b) cos(2A) 3. For an angle A it is known that 180 A 270 . If sin A (a) sin(2A) (b) cos(2A) 13 3 then find: 5 (c) tan(2A) 11 , find the following: 6 4. Which of the following is equivalent to 2sin50 cos50 ? 1) 2) sin 25 sin100 1) 2) cos100 cos 25 5. Which of the following is not equal to cos80 ? 1) 2) 1 sin 2 80 1 2sin 2 40 3) 2 cos 2 60 1 4) cos 2 40 sin 2 40 6. If cos B = .28, then cos2B is closest to 1) 0.56 3) -0.68 2) 0.14 4) -0.84 7. The value of cos130 is equal to 1) 3) 1 sin 2 130 2cos 2 260 1 2) 4) 2cos 65 1 2sin 2 65 There is a + or – in the front of each half angle identity because you must first determine what quadrant the half angle will be in. From there you can decide if the trig function (sine, cosine, or tangent) will be positive or negative. For example, if 270°<A<360°, Functions of the Half Angle 1 cos A 1 sin A 2 2 1 cos A 1 cos A 2 2 1 cos A 1 tan A 1 cos A 2 then ____< ½ A <____. 8. For an angle, A, it is known that 90 x 180 . If cos A 1 A 2 a) sin 1 A 2 b) cos 14 This is in Q__, so only ____ will be positive. 3 then find: 5 1 A 2 c) tan Exercise 4: For the angle A it is known that 180 A 270 and sin A 1 A 2 1 A 2 a) sin b) cos 5 . Find the following values: 13 2 1. Which of the following is equivalent to the expression 2 cos 60 1 ? 1) 2) 2. 1) 2) cos30 sin 30 3) 4) cos120 sin120 Which of the following is equal to sin10 ? 3) 4) 1 2sin 2 20 2sin 20 cos 20 2sin5 cos5 cos 2 5 sin 2 5 3. Which of the following is not equal to cos50 ? 1) 2) 1 sin 2 50 2 cos 2 100 1 4. If cos A 1) 2) 7 9 5 9 3) cos310 4) 1 2sin 2 25 1 , then cos(2 A) ? 3 2 3) 3 4) 3 3 15 1 A 2 c) tan 6. For an angle, B, it is known that 90 180 and cos B (a) sin(2B) 7. For an angle, , it is known that cos 5 . Find, in simplest form: 13 (b) cos(2B) 2 and 270 360 . Find, in simplest form: 4 (a) sin(2 ) (b) cos(2 ) 16 Lesson #94 Simplifying with the Identities A2.A.77 Apply the double-angle and half-angle formulas for trigonometric functions A2.A.67 Justify the Pythagorean identities A2.A.58 Know and apply the co-function and reciprocal relationships between trigonometric ratios The Sine and Cosine Functions of the Double Angle (from the reference sheet) Sin2A= Cos2A= Cos2A= Cos2A= We already did some simplifying in the second lesson of this unit. Now we will learn how to work with the double-angle formulas and Pythagorean identities when simplifying. We will still be using the reciprocal and co-function relationships as well. Remember, the double angle identities are given to you on the reference sheet, but the others are not. See how many of the identities you can remember. Use the process we learned to find the 2nd and 3rd Pythagorean Identities. Reciprocal Identities sec = csc = cot = Double Angle Identities Reminder, but will not be used this lesson sin 2 2sin cos Cofunction Identities cos 2 cos 2 sin 2 sin = cos = cos 2 2cos 2 1 tan = cot = cos 2 1 2sin 2 sec = csc = Quotient Identities Basic Pythagorean Identity tan = cot = Second Pythagorean Identity sin2θ = cos2θ = (with tanθ and secθ) 17 Third Pythagorean Identity (with cotθ and cscθ) General Hints Look for Double Angles (sin2A or cos2A). Whenever you see them, make a substitution. When you see the cosine of a double angle (cos2A), choose the substitution that will make other parts of that problem cancel. Whenever you see a reciprocal function or quotient function, make a substitution. Be on the lookout for squared trigonometric functions ( sin 2 x, cos2 x, etc. ) and be ready to use a Pythagorean Identity. This one is not automatic though. 1. Simplify the expression to a single trigonometric function: 3. Simplify the expression to a single trigonometric function : 1 sin 2 x . 4. Prove the Identity: 2 Simplify the expression to a single trigonometric function: 2. sin x cos x . cos x 2 sin (csc sin ) cos2 2 cos x . 1 cos 2 x 5. 6. 7. The expression (1 + cos x)(1 - cos x) is 8. equivalent to 2 (1) 1 (3) sin x 2 (2) sec x 9. 1 cos x csc x sin x Prove the identity: 2 2 (4) csc x Simplify: cos 2 x cos 2 tan 2 x 10. Prove the identity: 18 sin x cos x 1 cos x sin x cos x sin x 11. cos(2 A) sin 2 A is equivalent to which of the 12. following? (1) 1 2 (3) sin A 2 (2) cos A 2 (4) sin A The expression 2 cos is equivalent to sin 2 (1) cscø (3) cotø (2) secø (4) sinø 13. Simplify this complex fraction into a single trigonometric function: 14. 2 2 sin 2 x Express in simplest terms: cos x . 15. Simplify the expression to a single sin(2 x) trigonometric function: tan x 16. 24. Prove the identity: 17. 18. 19. 20. sin x cos x tan x csc x sec x cot x 1 Challenge: Express cos 2 A sin 2 A as a single cos A trigonometric function for all values of A for which the fraction is defined. 19 Lesson #95 – Solving Trigonometric Equations Graphically A2.A.68 Solve trigonometric equations for all values of the variable from 0° to 360° 2cos x 1 0 0 x 360 It helps to make a scale that goes by 90°. Set xscl on your calculator to match. Since our equations ask for the answers in degrees, this is the only time we graph in degree mode! You used the variable, y, in this equation just so that you could graph. Therefore, just the x values of the intersections are your answers. Hint: Find and label the intersections first before sketching the graph. 15cos 2 x 7 cos x 2 0 0 x 360 20 Note: Equations do not have to be set equal to zero. Sometimes it is easier to find the solutions when they are not. For example, set the next equation equal to 6. 3cos 2 x 7 cos x 6 0 0 x 360 . Note: On the regents, the problem will most likely not tell you to solve graphically. You have to REALIZE you can do so. In addition, even if a Regents problem tells you to solve algebraically, you can solve graphically and still receive ½ credit. On June 18th, I can’t tell you to graph – you must remember! 3sin(2 ) cos 0 2 cos 2 x 3sin x 3 0 21 0 x 360 Lesson #96 – Using Identities to Solve Trig. Equations A2.A.68 Solve trigonometric equations for all values of the variable from 0° to 360° A2.A.77 Apply the double-angle and half-angle formulas for trigonometric functions A2.A.67 Justify the Pythagorean identities The following equations are probably the most difficult to solve because they require a substitution. Remember, any trigonometric equation can be solved graphically. You will even receive half credit for this if the question specifically says to solve algebraically. Basic Pythagorean Identity: Trigonometric Identities for Equation Solving (2nd option) cos2A= (3rd option) Sin2A = cos2A= 0 A 360 sin(2 A) cos A 0 cos(2 A) 5cos A 3 0 22 0 A 360 cos 2 x 4sin x 4 0 0 x 360 0 360 3cos 2 2 cos(2 A) 3sin 2 A 6sin A 7 0 Try graphing this one too! 23