Use sum and difference formulas to verify identities.
Use sum and difference formulas for cosine
Finding the exact value of the sine, cosine, or tangent of an angle is often easier if we can rewrite the given angle in terms of two angles that have known trigonometric values. We can use the special angles, which we can review in the unit circle shown in Figure 2.
Figure 2. The Unit Circle
We will begin with the sum and difference formulas for cosine, so that we can find the cosine of a given angle if we can break it up into the sum or difference of two of the special angles.
First, we will prove the difference formula for cosines. Let’s consider two points on the unit circle. Point [latex]P[/latex] is at an angle [latex]\alpha [/latex] from the positive x-axis with coordinates [latex]\left(\cos \alpha ,\sin \alpha \right)[/latex] and point [latex]Q[/latex] is at an angle of [latex]\beta [/latex] from the positive x-axis with coordinates [latex]\left(\cos \beta ,\sin \beta \right)[/latex]. Note the measure of angle [latex]POQ[/latex] is [latex]\alpha -\beta [/latex].
Label two more points: [latex]A[/latex] at an angle of [latex]\left(\alpha -\beta \right)[/latex] from the positive x-axis with coordinates [latex]\left(\cos \left(\alpha -\beta \right),\sin \left(\alpha -\beta \right)\right)[/latex]; and point [latex]B[/latex] with coordinates [latex]\left(1,0\right)[/latex]. Triangle [latex]POQ[/latex] is a rotation of triangle [latex]AOB[/latex] and thus the distance from [latex]P[/latex] to [latex]Q[/latex] is the same as the distance from [latex]A[/latex] to [latex]B[/latex].
Figure 3. We can find the distance from [latex]P[/latex] to [latex]Q[/latex] using the distance formula.
How To: Given two angles, find the cosine of the difference between the angles.
Write the difference formula for cosine.
Substitute the values of the given angles into the formula.
Simplify.
Example 1: Finding the Exact Value Using the Formula for the Cosine of the Difference of Two Angles
Using the formula for the cosine of the difference of two angles, find the exact value of [latex]\cos \left(\frac{5\pi }{4}-\frac{\pi }{6}\right)[/latex].
Show Solution
Use the formula for the cosine of the difference of two angles. We have
Find the exact value of [latex]\cos \left(\frac{\pi }{3}-\frac{\pi }{4}\right)[/latex].
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[latex]\frac{\sqrt{2}+\sqrt{6}}{4}[/latex]
Example 2: Finding the Exact Value Using the Formula for the Sum of Two Angles for Cosine
Find the exact value of [latex]\cos \left({75}^{\circ }\right)[/latex].
Show Solution
As [latex]{75}^{\circ }={45}^{\circ }+{30}^{\circ }[/latex], we can evaluate [latex]\cos \left({75}^{\circ }\right)[/latex] as [latex]\cos \left({45}^{\circ }+{30}^{\circ }\right)[/latex]. Thus,
Example 4: Finding the Exact Value of an Expression Involving an Inverse Trigonometric Function
Find the exact value of [latex]\sin \left({\cos }^{-1}\frac{1}{2}+{\sin }^{-1}\frac{3}{5}\right)[/latex].
Show Solution
The pattern displayed in this problem is [latex]\sin \left(\alpha +\beta \right)[/latex]. Let [latex]\alpha ={\cos }^{-1}\frac{1}{2}[/latex] and [latex]\beta ={\sin }^{-1}\frac{3}{5}[/latex]. Then we can write
Finding exact values for the tangent of the sum or difference of two angles is a little more complicated, but again, it is a matter of recognizing the pattern.
Finding the sum of two angles formula for tangent involves taking quotient of the sum formulas for sine and cosine and simplifying. Recall, [latex]\tan x=\frac{\sin x}{\cos x},\cos x\ne 0[/latex].
We can use the sum and difference formulas to identify the sum or difference of angles when the ratio of sine, cosine, or tangent is provided for each of the individual angles. To do so, we construct what is called a reference triangle to help find each component of the sum and difference formulas.
To find [latex]\sin \left(\alpha +\beta \right)[/latex], we begin with [latex]\sin \alpha =\frac{3}{5}[/latex] and [latex]0<\alpha <\frac{\pi }{2}[/latex]. The side opposite [latex]\alpha [/latex] has length 3, the hypotenuse has length 5, and [latex]\alpha [/latex] is in the first quadrant. Using the Pythagorean Theorem, we can find the length of side [latex]a:[/latex]
Since [latex]\cos \beta =-\frac{5}{13}[/latex] and [latex]\pi <\beta <\frac{3\pi }{2}[/latex], the side adjacent to [latex]\beta [/latex] is [latex]-5[/latex], the hypotenuse is 13, and [latex]\beta [/latex] is in the third quadrant. Again, using the Pythagorean Theorem, we have
Since [latex]\beta [/latex] is in the third quadrant, [latex]a=-12[/latex].
Figure 5
The next step is finding the cosine of [latex]\alpha [/latex] and the sine of [latex]\beta [/latex]. The cosine of [latex]\alpha [/latex] is the adjacent side over the hypotenuse. We can find it from the triangle in Figure 5: [latex]\cos \alpha =\frac{4}{5}[/latex]. We can also find the sine of [latex]\beta [/latex] from the triangle in Figure 5, as opposite side over the hypotenuse: [latex]\sin \beta =-\frac{12}{13}[/latex]. Now we are ready to evaluate [latex]\sin \left(\alpha +\beta \right)[/latex].
A common mistake when addressing problems such as this one is that we may be tempted to think that [latex]\alpha [/latex] and [latex]\beta [/latex] are angles in the same triangle, which of course, they are not. Also note that
Now that we can find the sine, cosine, and tangent functions for the sums and differences of angles, we can use them to do the same for their cofunctions. You may recall that if the sum of two positive angles is [latex]\frac{\pi }{2}[/latex], those two angles are complements, and the sum of the two acute angles in a right triangle is [latex]\frac{\pi }{2}[/latex], so they are also complements. In Figure 6, notice that if one of the acute angles is labeled as [latex]\theta [/latex], then the other acute angle must be labeled [latex]\left(\frac{\pi }{2}-\theta \right)[/latex].
Figure 6. From these relationships, the cofunction identities are formed.
Notice also that [latex]\sin \theta =\cos \left(\frac{\pi }{2}-\theta \right):[/latex] opposite over hypotenuse. Thus, when two angles are complimentary, we can say that the sine of [latex]\theta [/latex] equals the cofunction of the complement of [latex]\theta [/latex]. Similarly, tangent and cotangent are cofunctions, and secant and cosecant are cofunctions.
A General Note: Cofunction Identities
The cofunction identities are summarized in the table below.
Write [latex]\sin \frac{\pi }{7}[/latex] in terms of its cofunction.
Show Solution
[latex]\cos \left(\frac{5\pi }{14}\right)[/latex]
Try It
Use sum and difference formulas to verify identities
Verifying an identity means demonstrating that the equation holds for all values of the variable. It helps to be very familiar with the identities or to have a list of them accessible while working the problems.
How To: Given an identity, verify using sum and difference formulas.
Begin with the expression on the side of the equal sign that appears most complex. Rewrite that expression until it matches the other side of the equal sign. Occasionally, we might have to alter both sides, but working on only one side is the most efficient.
Look for opportunities to use the sum and difference formulas.
Rewrite sums or differences of quotients as single quotients.
If the process becomes cumbersome, rewrite the expression in terms of sines and cosines.
We see that the identity is verified. In many cases, verifying tangent identities can successfully be accomplished by writing the tangent in terms of sine and cosine.
Try It
Verify the identity: [latex]\tan \left(\pi -\theta \right)=-\tan \theta [/latex].
Example 10: Using Sum and Difference Formulas to Solve an Application Problem
Let [latex]{L}_{1}[/latex] and [latex]{L}_{2}[/latex] denote two non-vertical intersecting lines, and let [latex]\theta [/latex] denote the acute angle between [latex]{L}_{1}[/latex] and [latex]{L}_{2}[/latex]. Show that
where [latex]{m}_{1}[/latex] and [latex]{m}_{2}[/latex] are the slopes of [latex]{L}_{1}[/latex] and [latex]{L}_{2}[/latex] respectively. (Hint: Use the fact that [latex]\tan {\theta }_{1}={m}_{1}[/latex] and [latex]\tan {\theta }_{2}={m}_{2}[/latex]. )
Figure 7
Show Solution
Using the difference formula for tangent, this problem does not seem as daunting as it might.
For a climbing wall, a guy-wire [latex]R[/latex] is attached 47 feet high on a vertical pole. Added support is provided by another guy-wire [latex]S[/latex] attached 40 feet above ground on the same pole. If the wires are attached to the ground 50 feet from the pole, find the angle [latex]\alpha [/latex] between the wires.
Show Solution
Let’s first summarize the information we can gather from the diagram. As only the sides adjacent to the right angle are known, we can use the tangent function. Notice that [latex]\tan \beta =\frac{47}{50}[/latex], and [latex]\tan \left(\beta -\alpha \right)=\frac{40}{50}=\frac{4}{5}[/latex]. We can then use difference formula for tangent.
Occasionally, when an application appears that includes a right triangle, we may think that solving is a matter of applying the Pythagorean Theorem. That may be partially true, but it depends on what the problem is asking and what information is given.
The sum formula for cosines states that the cosine of the sum of two angles equals the product of the cosines of the angles minus the product of the sines of the angles. The difference formula for cosines states that the cosine of the difference of two angles equals the product of the cosines of the angles plus the product of the sines of the angles.
The sum and difference formulas can be used to find the exact values of the sine, cosine, or tangent of an angle.
The sum formula for sines states that the sine of the sum of two angles equals the product of the sine of the first angle and cosine of the second angle plus the product of the cosine of the first angle and the sine of the second angle. The difference formula for sines states that the sine of the difference of two angles equals the product of the sine of the first angle and cosine of the second angle minus the product of the cosine of the first angle and the sine of the second angle.
The sum and difference formulas for sine and cosine can also be used for inverse trigonometric functions.
The sum formula for tangent states that the tangent of the sum of two angles equals the sum of the tangents of the angles divided by 1 minus the product of the tangents of the angles. The difference formula for tangent states that the tangent of the difference of two angles equals the difference of the tangents of the angles divided by 1 plus the product of the tangents of the angles.
The Pythagorean Theorem along with the sum and difference formulas can be used to find multiple sums and differences of angles.
The cofunction identities apply to complementary angles and pairs of reciprocal functions.
Sum and difference formulas are useful in verifying identities.
Application problems are often easier to solve by using sum and difference formulas.