verifying trigonometric identities section 5.1. objectives apply a pythagorean identity to solve a...
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Verifying Trigonometric Identities
Section 5.1
Objectives• Apply a Pythagorean identity to solve
a trigonometric equation. • Rewrite an expression in terms of
sines and cosines using definitions and identities.
• Verify a trigonometric identity.
Pythagorean Identities
1)(cos)(sin 22 xx
)(csc)(cot1 22 xx
)(sec1)(tan 22 xx
Reciprocal Identities
)csc(1
)sin(t
t
)sec(1
)cos(t
t
)cot(1
)tan(t
t
)sin(1
)csc(t
t
)cos(1
)sec(t
t
)tan(1
)cot(t
t
Quotient Identities
)cos()sin(
)tan(tt
t
)sin()cos(
)cot(tt
t
Even-Odd Identites
)sin()sin( tt
)cos()cos( tt
)tan()tan( tt
)csc()csc( tt
)sec()sec( tt
)cot()cot( tt
Verifying Trigonometric Identities
When verifying trigonometric identities, a main way to start is to change everything to be in terms of sine and cosine function before trying to simplify. In the next two problems, we can see the process of changing everything to be in terms of sines and cosines. In the first problem, we use the Pythagorean identity to find trigonometric function values. In the second, we verify a trigonometric identity.
Evaluate the following expressions using identities if and7
8)tan( x 0)sin( x
)sec(x
)csc(x
)cos(x
)sin(x )cot(x
)tan(x
continued on next slide
To solve this problem, we are going to rewrite the tangent function in terms of the sine and the cosine function. Once we have done that, we will use the Pythagorean identity to find the values of the sine and cosine functions. Once we have these, we can find the values of all of the other functions.
Evaluate the following expressions using identities if and7
8)tan( x 0)sin( x
)cos()sin(
)tan(xx
x
continued on next slide
Based on the definition of the tangent function:
78
)cos()sin(
xx
Thus
Evaluate the following expressions using identities if and7
8)tan( x 0)sin( x
1)(cos)(sin 22 xx
continued on next slide
We now have two equations with two unknowns. Such equations we can solve using substitution. We will solve the first equation for sin(x) and then substitute that into the second equation.
78
)cos()sin(
xx
Evaluate the following expressions using identities if and7
8)tan( x 0)sin( x
49)(cos49)(cos64
1)(cos)(cos4964
1)(cos)cos(78
22
22
22
xx
xx
xx
continued on next slide
Plug into second equation and solve for cos(x)
)cos(78
)sin( xx
Evaluate the following expressions using identities if and7
8)tan( x 0)sin( x
11349
)cos(
11349
)(cos
49)(cos113
2
2
x
x
x
continued on next slide
Since the tangent value is negative and the sine value is negative, the angle x must be in quadrant IV. Thus the cosine value is positive.
113
7)cos( x
Evaluate the following expressions using identities if and7
8)tan( x 0)sin( x
113
8)sin(
113
778
)sin(
)cos(78
)sin(
x
x
xx
continued on next slide
Now we plug this value in for cos(x) to find the sin(x)
Evaluate the following expressions using identities if and7
8)tan( x 0)sin( x
113
7)cos( x
113
8)sin( x
)cot(x 78
11371138
)cos()sin(
)tan(
xx
x
continued on next slide
Now we are ready to use the sine and cosine values to find the other trigonometric function values.
Evaluate the following expressions using identities if and7
8)tan( x 0)sin( x
113
7)cos( x
113
8)sin( x
87
1138
1137
)sin()cos(
)cot(
xx
x
continued on next slide
Now we are ready to use the sine and cosine values to find the other trigonometric function values.
Evaluate the following expressions using identities if and7
8)tan( x 0)sin( x
7113
11371
)cos(1
)sec(
x
x
8113
1138
1)sin(
1)csc(
x
x
113
7)cos( x
113
8)sin( x
Continuing with the final two trigonometric functions.
Verify the trigonometric identity.
)(cos1)(sec
)(tan2 22
2
xxx
When verifying a trigonometric identity, we work with one side of the equation until we make it look exactly like the other side of the equation. When working with the one side of the equation, we can multiply that side by 1 (or any fraction that is equal to 1), rearrange terms within the side we are working on, combine or take apart fractions that are on the side we are working on, and replace terms with equal terms. The one thing that we cannot do is move things from one side of the equation to the other.
For this example, since the left side is more complicated, we will work with that side trying to simplify it.
continued on next slide
Verify the trigonometric identity.
)(cos1)(sec
)(tan2 22
2
xxx
continued on next slide
We will start by changing all the functions on the left side to be in terms of sine and cosine functions.
1
)(cos1
)(cos)(sin
2
2
2
2
x
xx
Notice that since we are only working with the left side, we do not even write the right side. This will keep us from trying to do things to both sides of the equation.
Verify the trigonometric identity. )(cos1)(sec
)(tan2 22
2
xxx
continued on next slide
Our next step will be to get a common denominator for the two parts of the numerator. This will allow us to write the numerator as a singe fraction.
1
)(cos1
)(cos)(sin)(cos2
1
)(cos1
)(cos)(sin
)(cos)(cos2
2
2
22
2
2
2
2
2
x
xxx
x
xx
xx
In our next step we will simplify the complex fraction. Since one fraction is being divided by another fraction, we can multiply the fraction in the numerator of the large fraction by the multiplicative inverse of the fraction in the denominator of the large fraction.
Verify the trigonometric identity. )(cos1)(sec
)(tan2 22
2
xxx
continued on next slide
1)(sin)(cos2
11
)(cos)(cos
)(sin)(cos2
1)(cos
1)(cos
)(sin)(cos2
1
)(cos1
)(cos)(sin)(cos2
22
2
2
22
22
22
2
2
22
xx
xx
xx
xxxx
x
xxx
Next we should be reminded that what is left looks similar to the Pythagorean identity. The difference between the Pythagorean identity and what we have is the 2 in front of the cos2(x). We are going to rewrite this expression to change it from 2 cos2(x) to cos2(x). + cos2(x).
1)(sin)(cos)(cos
1)(sin)(cos2222
22
xxx
xx
Verify the trigonometric identity. )(cos1)(sec
)(tan2 22
2
xxx
1)(cos)(sin 22 xxPythagorean identity:
Now we can see that the encircled part matches one side of the Pythagorean identity. We can replace this part with what the Pythagorean identity says that it is equal to.
Thus we can replace the encircled part with 1.
)(cos
11)(cos2
2
x
x
Notice that we have now made what was on the left side of the equation look exactly like what was on the right side. This means that we have verified the identity.