3.1 Fundamental Identities

Transcription

1 Chapter. Trigonometric Identities and Equations. Fundamental Identities Introduction We now enter into the proof portion of trigonometry. Starting with the basic definitions of sine, cosine, and tangent, identities (or fundamental trigonometric equations) emerge. Students will learn how to prove certain identities, using other identities and definitions. Finally, students will be able solve trigonometric equations for theta, also using identities and definitions. Learning Objectives use the fundamental identities to prove other identities. apply the fundamental identities to values of θ and show that they are true. Quotient Identity In Chapter, the three fundamental trigonometric functions sine, cosine and tangent were introduced. All three functions can be defined in terms of a right triangle or the unit circle. 89

2 .. Fundamental Identities sinθ= opposite hypotenuse = y r = y = y ad jacent = hypotenuse = x r = x = x tanθ= opposite ad jacent = y x = sinθ The Quotient Identity is tanθ = sinθ. We see that this is true because tangent is equal to y x, in the unit circle. We know that y is equal to the sine values of θ and x is equal to the cosine values of θ. Substituting sinθ for y and for x and we have a new identity. Example : Use θ=45 to show that tanθ= sinθ holds true. Solution: Plugging in 45, we have: tan45 = sin45 cos45. Then, substitute each function with its actual value and simplify both sides. sin45 cos45 = = = = and we know that tan 45 =, so this is true. Example : Show that tan 90 is undefined using the Quotient Identity. Solution: tan90 = sin90 cos90 = 0, because you cannot divide by zero, the tangent at 90 is undefined. Reciprocal Identities Chapter also introduced us to the idea that the three fundamental reciprocal trigonometric functions are cosecant (csc), secant (sec) and cotangent (cot) and are defined as: cscθ= sinθ secθ= cotθ= tanθ If we apply the Quotient Identity to the reciprocal of tangent, an additional quotient is created: Example : Prove tanθ=sinθsecθ cotθ= tanθ = sinθ = sinθ Solution: First, you should change everything into sine and cosine. Feel free to work from either side, as long as the end result from both sides ends up being the same. tanθ=sinθ secθ = sinθ = sinθ Here, we end up with the Quotient Identity, which we know is true. Therefore, this identity is also true and we are finished. 90

3 Chapter. Trigonometric Identities and Equations Example 4: Given sinθ= 6 5 and θ is in the fourth quadrant, find secθ. Solution: Secant is the reciprocal of cosine, so we need to find the adjacent side. We are given the opposite side, 6 and the hypotenuse, 5. Because θ is in the fourth quadrant, cosine will be positive. From the Pythagorean Theorem, the third side is: ( 6 ) + b = 5 6+b = 5 b = 9 b= 9 From this we can now find = 9 5. Since secant is the reciprocal of cosine, secθ= 5 9, or Pythagorean Identity Using the fundamental trig functions, sine and cosine and some basic algebra can reveal some interesting trigonometric relationships. Note when a trig function such as sin θ is multiplied by itself, the mathematical convention is to write it as sin θ. (sinθ can be interpreted as the sine of the square of the angle, and is therefore avoided.) sin θ= y and cos θ= x or sin θ+cos θ= y + x = x +y r r r r r r Using the Pythagorean Theorem for the triangle above: x + y = r Then, divide both sides by r, x +y r = r =. So, because x +y =,sin θ+cos θ also equals. This is known as r r the Trigonometric Pythagorean Theorem or the Pythagorean Identity and is written sin θ+cos θ=. Alternative forms of the Theorem are: +cot θ=csc θ and tan θ+=sec θ. The second form is found by taking the original form and dividing each of the terms by sin θ, while the third form is found by dividing all the terms of the first by cos θ. Example 5: Use 0 to show that sin θ+cos θ= holds true. Solution: Plug in 0 and find the values of sin0 and cos0. sin 0 + cos 0 ( ) ( = ) Even and Odd Identities Functions are even or odd depending on how the end behavior of the graphical representation looks. For example, y = x is considered an even function because the ends of the parabola both point in the same direction and the parabola is symmetric about the y axis. y=x is considered an odd function for the opposite reason. The ends of 9

4 .. Fundamental Identities a cubic function point in opposite directions and therefore the parabola is not symmetric about the y axis. What about the trig functions? They do not have exponents to give us the even or odd clue (when the degree is even, a function is even, when the degree is odd, a function is odd). Even Function Odd Function y=( x) = x y=( x) = x Let s consider sine. Start with sin( x). Will it equal sinx or sinx? Plug in a couple of values to see. sin( 0 )=sin0 = = sin0 sin( 5 )=sin5 = = sin5 From this we see that sine is odd. Therefore, sin( x) = sinx, for any value of x. For cosine, we will plug in a couple of values to determine if it s even or odd. cos( 0 )=cos0 = = cos0 cos( 5 )=cos5 = = cos5 This tells us that the cosine is even. Therefore, cos( x) =, for any value of x. The other four trigonometric functions are as follows: tan( x)= tanx csc( x)= cscx sec( x)=secx cot( x)= cotx Notice that cosecant is odd like sine and secant is even like cosine. Example 6: If cos( x)= 4 and tan( x)= 7, find sinx. Solution: We know that sine is odd. Cosine is even, so = 4. Tangent is odd, so tanx= 7. Therefore, sine is positive and sinx= 7 4. Cofunction Identities Recall that two angles are complementary if their sum is 90. In every triangle, the sum of the interior angles is 80 and the right angle has a measure of 90. Therefore, the two remaining acute angles of the triangle have a sum equal to 90, and are complementary. Let s explore this concept to identify the relationship between a function of one angle and the function of its complement in any right triangle, or the cofunction identities. A cofunction is a pair of trigonometric functions that are equal when the variable in one function is the complement in the other. In ABC, C is a right angle, A and B are complementary. 9

5 Chapter. Trigonometric Identities and Equations Chapter introduced the cofunction identities (section.8) and because A and B are complementary, it was found that sina=cosb,cosa=sinb,tana=cotb,cota=tanb,csca=secb and seca=cscb. For each of the above A= π B. To generalize, sin ( π θ) = and cos ( π θ) = sinθ,tan ( π θ) = cotθ and cot ( π θ) = tanθ,csc ( π θ) = secθ and sec ( π θ) = cscθ. The following graph represents two complete cycles of y=sinx and y=. Notice that a phase shift of π on y=, would make these graphs exactly the same. These cofunction identities hold true for all real numbers for which both sides of the equation are defined. Example 7: Use the cofunction identities to evaluate each of the following expressions: a. If tan ( π θ) = 4.6 determine cotθ b. If sinθ=0.9 determine cos ( π θ). Solution: a. tan ( π θ) = cotθ therefore cotθ= 4.6 b. cos ( π θ) = sinθ therefore cos ( π θ) = 0.9 Example 8: Show sin ( π x) = cos( x) is true. Solution: Using the identities we have derived in this section, sin ( π x) =, and we know that cosine is an even function so cos( x)=. Therefore, each side is equal to and thus equal to each other. Points to Consider Why do you think secant is even like cosine? How could you show that tangent is odd? 9

6 .. Fundamental Identities Review Questions. Use the Quotient Identity to show that the tan 70 is undefined.. If cos ( π x) = 4 5, find sin( x).. If tan( x)= 5 5 and sinx=, find. 4. Simplify secxcos ( π x). 5. Verify sin θ+cos θ= using: a. the sides 5,, and of a right triangle, in the first quadrant b. the ratios from a triangle 6. Prove +tan θ=sec θ using the Pythagorean Identity 7. If cscz= and cosz= 7, find cotz. 8. Factor: a. sin θ cos θ b. sin θ+6sinθ+8 9. Simplify sin4 θ cos 4 θ using the trig identities sin θ cos θ 0. Rewrite secx so that it is only in terms of cosine. Simplify completely.. Prove that tangent is an odd function. Review Answers 94. tan70 = sin70 0, you cannot divide by zero, therefore tan70 is undefined.. If cos ( π x) = 4 5, then, by the cofunction identities, sinx= 4 5. Because sine is odd, sin( x)= If tan( x)= 5, then tanx= 5. Because sinx= 5, cosine is also negative, so =. 4. Use the reciprocal and cofunction identities to simplify ( π ) secxcos x cos70 = sinx sinx tanx 5. (a) Using the sides 5,, and and in the first quadrant, it doesn t really matter which is cosine or sine. So, sin θ+cos θ= becomes ( ) 5 + ( ) =. Simplifying, we get: =, and finally 69 =. (b) sin θ+cos θ= becomes ( ) ( ) + =. Simplifying we get: = and 4 4 =. 6. To prove tan θ+=sec θ, first use sinθ = tanθ and change sec θ= cos θ. 7. If cscz= 7 8 tan θ+=sec θ sin θ cos θ + = cos θ sin θ cos θ + cos θ cos θ = cos θ sin θ+cos θ= 5 8 and cosz= 7, then sinz= 7 and tanz= Therefore cotz= 8.

7 Chapter. Trigonometric Identities and Equations 8. (a) Factor sin θ cos θ using the difference of squares. sin θ cos θ=(sinθ+)(sinθ ) (b) sin θ+6 sinθ+8=(sinθ+4)(sinθ+) 9. You will need to factor and use the sin θ+cos θ= identity. sin 4 θ cos 4 θ sin θ cos θ = (sin θ cos θ)(sin θ+cos θ) sin θ cos θ = sin θ+cos θ 0. To rewrite secx so it is only in terms of cosine, start with changing secant to cosine. secx = = = Now, simplify the denominator. Multiply by the reciprocal = = = cos x. The easiest way to prove that tangent is odd to break it down, using the Quotient Identity. tan( x)= sin( x) cos( x) = sinx = tanx from this statement, we need to show that tan( x)= tanx because sin( x)= sinx and cos( x)= 95