# 9.2 Sum and difference identities  (Page 5/6)

 Page 5 / 6

Verify the identity: $\text{\hspace{0.17em}}\mathrm{tan}\left(\pi -\theta \right)=-\mathrm{tan}\text{\hspace{0.17em}}\theta .$

$\begin{array}{ccc}\hfill \mathrm{tan}\left(\pi -\theta \right)& =& \frac{\mathrm{tan}\left(\pi \right)-\mathrm{tan}\text{\hspace{0.17em}}\theta }{1+\mathrm{tan}\left(\pi \right)\mathrm{tan}\theta }\hfill \\ & =& \frac{0-\mathrm{tan}\text{\hspace{0.17em}}\theta }{1+0\cdot \mathrm{tan}\text{\hspace{0.17em}}\theta }\hfill \\ & =& -\mathrm{tan}\text{\hspace{0.17em}}\theta \hfill \end{array}$

## Using sum and difference formulas to solve an application problem

Let $\text{\hspace{0.17em}}{L}_{1}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{L}_{2}\text{\hspace{0.17em}}$ denote two non-vertical intersecting lines, and let $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ denote the acute angle between $\text{\hspace{0.17em}}{L}_{1}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{L}_{2}.\text{\hspace{0.17em}}$ See [link] . Show that

$\mathrm{tan}\text{\hspace{0.17em}}\theta =\frac{{m}_{2}-{m}_{1}}{1+{m}_{1}{m}_{2}}$

where $\text{\hspace{0.17em}}{m}_{1}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{m}_{2}\text{\hspace{0.17em}}$ are the slopes of $\text{\hspace{0.17em}}{L}_{1}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{L}_{2}\text{\hspace{0.17em}}$ respectively. ( Hint: Use the fact that $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}{\theta }_{1}={m}_{1}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}{\theta }_{2}={m}_{2}.$ )

Using the difference formula for tangent, this problem does not seem as daunting as it might.

$\begin{array}{ccc}\hfill \mathrm{tan}\text{\hspace{0.17em}}\theta & =& \mathrm{tan}\left({\theta }_{2}-{\theta }_{1}\right)\hfill \\ & =& \frac{\mathrm{tan}\text{\hspace{0.17em}}{\theta }_{2}-\mathrm{tan}\text{\hspace{0.17em}}{\theta }_{1}}{1+\mathrm{tan}\text{\hspace{0.17em}}{\theta }_{1}\mathrm{tan}\text{\hspace{0.17em}}{\theta }_{2}}\hfill \\ & =& \frac{{m}_{2}-{m}_{1}}{1+{m}_{1}{m}_{2}}\hfill \end{array}$

## Investigating a guy-wire problem

For a climbing wall, a guy-wire $\text{\hspace{0.17em}}R\text{\hspace{0.17em}}$ is attached 47 feet high on a vertical pole. Added support is provided by another guy-wire $\text{\hspace{0.17em}}S\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ between the wires. See [link] .

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 $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\beta =\frac{47}{50},$ and $\text{\hspace{0.17em}}\mathrm{tan}\left(\beta -\alpha \right)=\frac{40}{50}=\frac{4}{5}.\text{\hspace{0.17em}}$ We can then use difference formula for tangent.

$\mathrm{tan}\left(\beta -\alpha \right)=\frac{\mathrm{tan}\text{\hspace{0.17em}}\beta -\mathrm{tan}\text{\hspace{0.17em}}\alpha }{1+\mathrm{tan}\text{\hspace{0.17em}}\beta \mathrm{tan}\text{\hspace{0.17em}}\alpha }$

Now, substituting the values we know into the formula, we have

$\begin{array}{ccc}\hfill \frac{4}{5}& =& \frac{\frac{47}{50}-\mathrm{tan}\text{\hspace{0.17em}}\alpha }{1+\frac{47}{50}\mathrm{tan}\text{\hspace{0.17em}}\alpha }\hfill \\ \hfill 4\left(1+\frac{47}{50}\mathrm{tan}\text{\hspace{0.17em}}\alpha \right)& =& 5\left(\frac{47}{50}-\mathrm{tan}\text{\hspace{0.17em}}\alpha \right)\hfill \end{array}$

Use the distributive property, and then simplify the functions.

$\begin{array}{ccc}\hfill 4\left(1\right)+4\left(\frac{47}{50}\right)\mathrm{tan}\text{\hspace{0.17em}}\alpha & =& 5\left(\frac{47}{50}\right)-5\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\alpha \hfill \\ \hfill 4+3.76\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\alpha & =& 4.7-5\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\alpha \hfill \\ \hfill 5\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\alpha +3.76\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\alpha & =& 0.7\hfill \\ \hfill 8.76\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\alpha & =& 0.7\hfill \\ \hfill \mathrm{tan}\text{\hspace{0.17em}}\alpha & \approx & 0.07991\hfill \\ \hfill {\mathrm{tan}}^{-1}\left(0.07991\right)& \approx & .079741\hfill \end{array}$

Now we can calculate the angle in degrees.

$\alpha \approx 0.079741\left(\frac{180}{\pi }\right)\approx 4.57°$

Access these online resources for additional instruction and practice with sum and difference identities.

## Key equations

 Sum Formula for Cosine $\mathrm{cos}\left(\alpha +\beta \right)=\mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta -\mathrm{sin}\text{\hspace{0.17em}}\alpha \mathrm{sin}\text{\hspace{0.17em}}\beta$ Difference Formula for Cosine $\mathrm{cos}\left(\alpha -\beta \right)=\mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta +\mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\beta$ Sum Formula for Sine $\mathrm{sin}\left(\alpha +\beta \right)=\mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta +\mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\beta$ Difference Formula for Sine $\mathrm{sin}\left(\alpha -\beta \right)=\mathrm{sin}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\beta -\mathrm{cos}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\beta$ Sum Formula for Tangent $\mathrm{tan}\left(\alpha +\beta \right)=\frac{\mathrm{tan}\text{\hspace{0.17em}}\alpha +\mathrm{tan}\text{\hspace{0.17em}}\beta }{1-\mathrm{tan}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\beta }$ Difference Formula for Tangent $\mathrm{tan}\left(\alpha -\beta \right)=\frac{\mathrm{tan}\text{\hspace{0.17em}}\alpha -\mathrm{tan}\text{\hspace{0.17em}}\beta }{1+\mathrm{tan}\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\beta }$ Cofunction identities $\begin{array}{ccc}\hfill \mathrm{sin}\text{\hspace{0.17em}}\theta & =& \mathrm{cos}\left(\frac{\pi }{2}-\theta \right)\hfill \\ \hfill \mathrm{cos}\text{\hspace{0.17em}}\theta & =& \mathrm{sin}\left(\frac{\pi }{2}-\theta \right)\hfill \\ \hfill \mathrm{tan}\text{\hspace{0.17em}}\theta & =& \mathrm{cot}\left(\frac{\pi }{2}-\theta \right)\hfill \\ \hfill \mathrm{cot}\text{\hspace{0.17em}}\theta & =& \mathrm{tan}\left(\frac{\pi }{2}-\theta \right)\hfill \\ \hfill \mathrm{sec}\text{\hspace{0.17em}}\theta & =& \mathrm{csc}\left(\frac{\pi }{2}-\theta \right)\hfill \\ \hfill \mathrm{csc}\text{\hspace{0.17em}}\theta & =& \mathrm{sec}\left(\frac{\pi }{2}-\theta \right)\hfill \end{array}$

## Key concepts

• 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. See [link] and [link] .
• 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. See [link] .
• The sum and difference formulas for sine and cosine can also be used for inverse trigonometric functions. See [link] .
• 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. See [link] .
• The Pythagorean Theorem along with the sum and difference formulas can be used to find multiple sums and differences of angles. See [link] .
• The cofunction identities apply to complementary angles and pairs of reciprocal functions. See [link] .
• Sum and difference formulas are useful in verifying identities. See [link] and [link] .
• Application problems are often easier to solve by using sum and difference formulas. See [link] and [link] .

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