Module 6 Lesson 4 Using Trigonometric Identities -­‐ Part I You might recall from Algebra that an identity is a true statement. Here are a few examples. No matter what value is placed in for ‘x’, the equations will always hold true. 3( x − 6) = 3 x − 18
2 x − 5 x + 4 = 3 x − 2(3 x − 2) x=x
Trigonometric Identities use the same concept. No matter what value is plugged in for the angle measure, the equation will hold true. The Trigonometric Identities fall into a few different categories. Quotient Identities-­‐ There are two quotient identities. Recall that a quotient is the answer to a division problem. What happens when we divide sine by cosine? length of opposite side
length of adjacent side
Remember: sin x =
and cos x =
length of hypotenuse
length of hypotenuse
So if we divide those: length of opposite side length of adjacent side
÷
length of hypotenuse
length of hypotenuse
Division is multiplication by the reciprocal: length of opposite side length of hypotenuse
×
length of hypotenuse length of adjacent side
The ‘length of hypotenuse’ terms cancel out: length of opposite side length of hypotenuse
length of opposite side
×
=
length of hypotenuse length of adjacent side length of adjacent side
length of opposite side
And what is: ? You’ll recognize that as the Tangent function. length of adjacent side
1
We just found that: tan( x ) =
sin( x )
cos( x )
Using similar division, we can divide cosine by sine and find: cot( x) =
cos( x )
sin( x )
(We could have also found that result using reciprocals) So our two Quotient Identities are: sin( x )
cos( x )
tan( x ) =
cot( x) =
sin( x )
cos( x )
Pythagorean-­‐ The Pythagorean Identities get their name from the Pythagorean Theorem. Think back to what we’ve learned so far in Trigonometry: xo
Consider this angle, x. Let’s drop a perpendicular line from where the angle touches the unit circle down to the x-­‐axis. By doing so, we form a right triangle: 2
B
c
a
30o
b
A
Remember that in the Unit Circle, the hypotenuse is equal to the radius of the circle, which is 1. The horizontal component of the triangle is the x-­‐coordinate, which we learned was the value for the cosine of the angle. The vertical component of the triangle is the y-­‐coordinate, which we learned was the value for the sine of the angle. We now draw this picture: B
1
sin x
30o o
x
A
cos x
We can now use the Pythagorean Theorem to write this equation: sin 2 ( x) + cos2 ( x) = 1 This is our first Pythagorean Identity. It holds true no matter what angle value we put in for x. There are two other Pythagorean Identities. The first is formed by taking our original identity and dividing each term by sin 2 ( x ) : 3
sin 2 ( x ) + cos2 ( x ) = 1
sin 2 ( x ) cos2 ( x )
1
+
=
2
2
sin ( x ) sin ( x ) sin 2 ( x ) 1 + cot 2 ( x ) = csc 2 ( x )
Remember that cos x
1
= cot x , and = csc x sin x
sin x
The second is formed by taking our original identity and dividing each term by cos2 ( x ) : sin 2 ( x ) + cos2 ( x ) = 1
sin x
1
sin 2 ( x ) cos2 ( x )
1
Remember that = tan x , and = sec x +
=
2
2
2
cos x
cos x
cos ( x ) cos ( x ) cos ( x )
tan 2 ( x ) + 1 = sec 2 ( x )
So our three Pythagorean Identities are: sin 2 ( x ) + cos2 ( x ) = 1
1 + cot 2 ( x ) = csc 2 ( x ) tan 2 ( x ) + 1 = sec2 ( x )
Reciprocal Identities-­‐ We learned these earlier in Trigonometry, but we didn’t call them identities. Now, we can officially do that. Here are the six Reciprocal Identities: 1
1
sin( x ) =
csc( x ) =
csc( x )
sin( x )
1
1
cos( x ) =
sec( x ) =
sec( x )
cos( x )
1
1
tan( x ) =
cot( x ) =
cot( x )
tan( x )
Again, these hold true for every value of ‘x’. 4
Using the Identities-­‐ You can use the identities above to find values of different trigonometric functions of an angle if you already know one function. For example, let’s say you know the sine of an angle and you want to find the cosine of that angle. You could use the identity: sin 2 ( x) + cos2 ( x) = 1 to do so. What if you know the sine of an angle, and you want to know the tangent? You simply use sin 2 ( x) + cos2 ( x) = 1 to find the cosine first. Then once you know both the sine and sin x
cosine, you use this quotient identity: = tan x to find the value for tangent. cos x
What if you know the sine of an angle and want to find the secant? You first find the cosine with sin 2 ( x) + cos2 ( x) = 1, and then take the reciprocal of that answer – 1
according to the reciprocal identity: sec( x ) =
cos( x )
What if you have the sine of an angle and you need to find the cosecant? Simply take 1
the reciprocal, because our reciprocal identity csc( x ) =
tells us to. sin( x )
What if you have the sine of an angle and you need to find the cotangent? You have some options here. You could find the tangent as described above, and then take its 1
reciprocal since: cot( x ) =
. tan( x)
You could also take the reciprocal of sine first to find cosecant, and then use the Pythagorean Identity, 1 + cot 2 ( x) = csc2 ( x) , to find cotangent. cos( x )
Or you could find the cosine using sin 2 ( x) + cos2 ( x) = 1 , and then take cot( x ) =
sin( x )
according to our quotient identity. Using similar methods and these identities, you can go from any trigonometric function to any other. I like to use this picture to help figure the path: 5
SIN
COS
Rec.
Rec.
CSC
SEC
Q
COT
Rec.
TAN
I call this the Hexagon of Doom….because that’s the name we came up with in one of my math classes. The Hexagon helps you picture the path from one trig function to the next. Dark double-­‐headed arrows represent a Pythagorean relationship between those two trigonometric functions. The letters “Rec.” represent a reciprocal relationship between those two trigonometric functions. The capital Q in the middle represents ‘Quotient’ identities. It’s there to let you know that you can combine sine and cosine to find either tangent or cotangent. Example: Say you know the cosecant of an angle. How can you find the cosine of that angle? Answer: Consider the path around the Hexagon. We’d first take the reciprocal of cosecant to find sine, and then use the appropriate Pythagorean Identity to find cosine. Important-­‐ 6
The thing you have to remember is this. When you use these identities, you will, at times, be finding the square root of a value. How will you know when to use the positive square root and when to use the negative? The answer you use depends on the Quadrant your angle is in. We learned in our first Trigonometry Unit which Trigonometric Functions are positive in which quadrant. (A Smart Trig Class). This mnemonic should help guide you in finding the correct sign. Example 1: Given: sin x = 0.84 and that angle x is in quadrant II. a) Find csc x b) Find cos x c) Find tan x a) Find csc x 1
csc x =
sin x
1
Cosecant is the reciprocal of sine: csc x =
0.84
csc x = 1.190
b) We use the Pythagorean Identity to find cosine: sin 2 ( x ) + cos2 ( x ) = 1
cos2 ( x ) = 1 − sin 2 ( x )
cos2 ( x ) = 1 − (0.84) 2 When we get to this point, we need to square root both sides: cos2 ( x ) = 1 − 0.7056
cos2 ( x ) = 0.2944
cos2 ( x ) = 0.2944
cos2 ( x ) = ± 0.2944
Notice the plus/minus appears when you take the square root cos2 ( x ) = ± 0.2944 So, do we use the positive or negative? cos x = ±0.5426
Since we’re in Quadrant II, cosine values are negative. Therefore: cos x = -­‐0.5426 7
c) To find Tangent, we now have a couple of options. The best one is to use our new cosine value and our given sine value and the appropriate quotient identity: tan( x ) =
sin( x )
0.84
=
= −1.548 cos( x ) −0.5426
The sign of this answer is negative, as it should be for tangent values in Quadrant II. 8