Unit 2

Trigonometry

Certainly not all equations containing trigonometric ratios are identities. When given an equation, you can check if it is an identity using two different procedures.

Verifying an identity can be used to show an equation is not an identity. Verifying cannot be used to guarantee an equation is an identity.

Proving an identity can be used to guarantee an equation is an identity. Proofs will be covered in part 2.4C.

Verifying an Identity

An identity can be verified numerically by substituting a value for the variable. If the values on each side of the equation are the same, the equation may be an identity. If the two sides are not equal, the equation cannot be an identity.

An identity can also be verified graphically by graphing the functions that correspond to the left and right sides of the equation. If the graphs look the same, the equation may be an identity. If the graphs are different, the equation is not an identity.

Consider the equation «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfrac»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»cot«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mi»sin«/mi»«mi»x«/mi»«/mrow»«/mstyle»«/math».

a.
Determine any non-permissible values for the equation.
b.
Verify the equation numerically using «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mfrac»«mo»§#960;«/mo»«mn»6«/mn»«/mfrac»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mfrac»«mo»§#960;«/mo»«mn»3«/mn»«/mfrac»«/mrow»«/mstyle»«/math».
c
Verify the equation graphically.
a.
Non-permissible values can be found by converting expressions to sines and cosines, and looking at where denominators are equal to zero.

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»cot«/mi»«mi»x«/mi»«mo»=«/mo»«mfrac»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»sin«/mi»«mi»x«/mi»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math» , so the equation «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfrac»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»cot«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mi»sin«/mi»«mi»x«/mi»«/mrow»«/mstyle»«/math» can be rewritten as

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfrac»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«mfenced»«mfrac»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»sin«/mi»«mi»x«/mi»«/mrow»«/mfrac»«/mfenced»«/mfrac»«mo»=«/mo»«mi»sin«/mi»«mi»x«/mi»«/mrow»«/mstyle»«/math».

Both «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»cos«/mi»«mi»x«/mi»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»sin«/mi»«mi»x«/mi»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math» will make the left side of the equation undefined.

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»cos«/mi»«mi»x«/mi»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math» when «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mfrac»«mo»§#960;«/mo»«mn»2«/mn»«/mfrac»«mo»+«/mo»«mi»n«/mi»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mi»n«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»I«/mi»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»sin«/mi»«mi»x«/mi»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math» when «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mi»n«/mi»«mo»§#960;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mi»n«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»I«/mi»«/mrow»«/mstyle»«/math». These expressions can be combined to give the restrictions on «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math».

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8800;«/mo»«mfrac»«mrow»«mi»n«/mi»«mo»§#960;«/mo»«/mrow»«mn»2«/mn»«/mfrac»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mi»n«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»I«/mi»«/mrow»«/mstyle»«/math»

b.
Left Side Right Side
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right¨»«mtr»«mtd»«mfrac»«mrow»«mi»cos«/mi»«mfrac»«mo»§#960;«/mo»«mn»6«/mn»«/mfrac»«/mrow»«mrow»«mi»cot«/mi»«mfrac»«mo»§#960;«/mo»«mn»6«/mn»«/mfrac»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mfenced»«mfrac»«msqrt»«mn»3«/mn»«/msqrt»«mn»2«/mn»«/mfrac»«/mfenced»«mfenced»«msqrt»«mn»3«/mn»«/msqrt»«/mfenced»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mo»=«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/mstyle»«/math» «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»sin«/mi»«mfrac»«mo»§#960;«/mo»«mn»6«/mn»«/mfrac»«mo»=«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mrow»«/mstyle»«/math»

The left side equals the right side, so the equation has been verified for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mfrac»«mo»§#960;«/mo»«mn»6«/mn»«/mfrac»«/mrow»«/mstyle»«/math». This result means the given equation is a potential identity.

Left Side Right Side
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right¨»«mtr»«mtd»«mfrac»«mrow»«mi»cos«/mi»«mfrac»«mo»§#960;«/mo»«mn»3«/mn»«/mfrac»«/mrow»«mrow»«mi»cot«/mi»«mfrac»«mo»§#960;«/mo»«mn»3«/mn»«/mfrac»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mfenced»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mfenced»«mfenced»«mfrac»«mn»1«/mn»«msqrt»«mn»3«/mn»«/msqrt»«/mfrac»«/mfenced»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mo»=«/mo»«mfrac»«msqrt»«mn»3«/mn»«/msqrt»«mn»2«/mn»«/mfrac»«/mtd»«/mtr»«/mtable»«/mstyle»«/math» «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»sin«/mi»«mfrac»«mo»§#960;«/mo»«mn»3«/mn»«/mfrac»«mo»=«/mo»«mfrac»«msqrt»«mn»3«/mn»«/msqrt»«mn»2«/mn»«/mfrac»«/mrow»«/mstyle»«/math»

The left side equals the right side, so the equation has been verified for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mfrac»«mo»§#960;«/mo»«mn»3«/mn»«/mfrac»«/mrow»«/mstyle»«/math». This result also means the given equation is a potential identity.

c.
Graph the functions corresponding to each side of the equation to verify.

The two graphs look the same. This is a strong clue the equation is an identity, but it is not a guarantee.


Consider the equation «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»sec«/mi»«mi»x«/mi»«mi»tan«/mi»«mi»x«/mi»«mo»=«/mo»«mi»csc«/mi»«mi»x«/mi»«/mrow»«/mstyle»«/math».
a.
Determine any non-permissible values for the equation.
b.
Verify the equation numerically using «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»45«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»60«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math».
c.
Verify the equation graphically.
a.
Rewriting the equation using sines and cosines can help determine non-permissible values.

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable»«mtr»«mtd»«mi»sec«/mi»«mi»x«/mi»«mi»tan«/mi»«mi»x«/mi»«mo»=«/mo»«mi»csc«/mi»«mi»x«/mi»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mn»1«/mn»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»§#8226;«/mo»«mfrac»«mrow»«mi»sin«/mi»«mi»x«/mi»«/mrow»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/mfrac»«mo»=«/mo»«mfrac»«mn»1«/mn»«mrow»«mi»sin«/mi»«mi»x«/mi»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»

Both «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»sin«/mi»«mi»x«/mi»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»cos«/mi»«mi»x«/mi»«/mrow»«/mstyle»«/math» will make a denominator equal to zero, so «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8800;«/mo»«mn»90«/mn»«mo»§#176;«/mo»«mi»n«/mi»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mi»n«/mi»«mo»§#8712;«/mo»«mi mathvariant=¨normal¨»I«/mi»«/mrow»«/mstyle»«/math».

b.
Left Side Right Side
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right¨»«mtr»«mtd»«mi»sec«/mi»«mn»45«/mn»«mo»§#176;«/mo»«mo»§#8226;«/mo»«mi»tan«/mi»«mn»45«/mn»«mo»§#176;«/mo»«mo»=«/mo»«msqrt»«mn»2«/mn»«/msqrt»«mo»§#8226;«/mo»«mn»1«/mn»«/mtd»«/mtr»«mtr»«mtd»«mo»=«/mo»«msqrt»«mn»2«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«/mstyle»«/math» «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»csc«/mi»«mn»45«/mn»«mo»§#176;«/mo»«mo»=«/mo»«msqrt»«mn»2«/mn»«/msqrt»«/mrow»«/mstyle»«/math»

The left side equals the right side, so the equation has been verified for «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»45«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math». This result means the given equation is a potential identity.

Left Side Right Side
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right¨»«mtr»«mtd»«mi»sec«/mi»«mn»60«/mn»«mo»§#176;«/mo»«mo»§#8226;«/mo»«mi»tan«/mi»«mn»60«/mn»«mo»§#176;«/mo»«mo»=«/mo»«mn»2«/mn»«mo»§#8226;«/mo»«msqrt»«mn»3«/mn»«/msqrt»«/mtd»«/mtr»«mtr»«mtd»«mo»=«/mo»«mn»2«/mn»«msqrt»«mn»3«/mn»«/msqrt»«/mtd»«/mtr»«/mtable»«/mstyle»«/math» «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»csc«/mi»«mn»60«/mn»«mo»§#176;«/mo»«mo»=«/mo»«mfrac»«mn»2«/mn»«msqrt»«mn»3«/mn»«/msqrt»«/mfrac»«/mrow»«/mstyle»«/math»

The left side does not equal the right side, so the equation is not an identity.
c.
The graphs do not match, so the equation is not an identity.
In Example 3, the numerical verification using «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»45«/mn»«mo»§#176;«/mo»«/mrow»«/mstyle»«/math» made the two sides of the equation equal. This suggested the equation might be an identity, but further verifications showed that it was not. The more verifications that are true, the more likely an equation is an identity. However, you cannot guarantee a trigonometric equation is an identity by using only verifications. Sometimes, the values chosen for verification end up simply being solutions to the given equation.