In this problem, we will derive formulas for calculating the cosine and sine of the sum and difference of angles whose sum is less than $90^{\circ}$.

![](https://cdn.mathpix.com/cropped/2024_05_01_e988dd467f3d6c2bf99fg-07.jpg?height=494&width=896&top_left_y=530&top_left_x=652)

In the figure above, the segment $X Z$ measures 1. Remembering that the sine of an angle in a right triangle is the quotient of the opposite leg and the hypotenuse, and that the cosine is the quotient of the adjacent leg and the hypotenuse, we can write: $\cos B=Z T / 1$, $\operatorname{sen} B=T X / 1, \cos A=Z L / Z T$, sen $A=K X / T X$ and $\cos (A+B)=Y Z / 1$. Since $Y Z=L Z-$ $L Y$, we have:

$$
\begin{aligned}
Y Z & =L Z-L Y \\
\cos (A+B) & =\cos A \cdot Z T-X K \\
& =\cos A \cdot \cos B-\operatorname{sen} A \cdot T X \\
& =\cos A \cdot \cos B-\operatorname{sen} A \cdot \operatorname{sen} B
\end{aligned}
$$

a) Use the figure above to determine the formula for $\operatorname{sen}(A+B)$ in terms of sen $A$, $\operatorname{sen} B, \cos A$ and $\cos B$.

b) Consider the following drawing:

![](https://cdn.mathpix.com/cropped/2024_05_01_e988dd467f3d6c2bf99fg-07.jpg?height=486&width=516&top_left_y=1856&top_left_x=847)

Using the ideas presented in the statement, find the expressions for $\cos (A-B)$ and $\operatorname{sen}(A-B)$ using the trigonometric functions of $A$ and $B$.
Solution

a) We have

$$
\operatorname{sen}(A+B)=\frac{X Y}{1}=K T+T L=\cos A \cdot \operatorname{sen} B+\operatorname{sen} A \cdot \cos B
$$

b) Using the trigonometric functions in the given right triangles, we have the following measures:

![](https://cdn.mathpix.com/cropped/2024_05_01_e988dd467f3d6c2bf99fg-08.jpg?height=700&width=1089&top_left_y=813&top_left_x=370)

Then we have

$$
\begin{aligned}
\cos (A-B) & =\frac{Y Z}{1} \\
& =K T+T L \\
& =\operatorname{sen} A \cdot \operatorname{sen} B+\cos A \cdot \cos B
\end{aligned}
$$

$$
\begin{aligned}
\operatorname{sen}(A-B) & =\frac{X Y}{1} \\
& =L Z-X K \\
& =\operatorname{sen} A \cdot \cos B-\cos A \cdot \operatorname{sen} B
\end{aligned}
$$