Answer:
1) 67.38°
[tex]\textsf{2)} \quad \cos^{-1}\left(\dfrac{5}{13}\right)[/tex]
Step-by-step explanation:
To find the measure of angle A in the given right triangle, we can use the cosine trigonometric ratio:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Cosine trigonometric ratio}}\\\\\sf \cos(\theta)=\dfrac{A}{H}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$\theta$ is the angle.}\\\phantom{ww}\bullet\;\textsf{$A$ is the side adjacent the angle.}\\\phantom{ww}\bullet\;\textsf{$H$ is the hypotenuse (the side opposite the right angle).}\end{array}}[/tex]
In this case, the side adjacent to angle A is AB, and the hypotenuse of the triangle is AC, so:
[tex]\cos A=\dfrac{AB}{AC}[/tex]
Given that AB = 5 and AC = 13, then:
[tex]\cos A=\dfrac{5}{13}[/tex]
Solve for angle A by taking the inverse cosine of both sides:
[tex]\cos^{-1}\left(\cos A\right)=\cos^{-1}\left(\dfrac{5}{13}\right)\\\\\\A=\cos^{-1}\left(\dfrac{5}{13}\right)\\\\\\A=67.38013505...^{\circ}\\\\\\A=67.38^{\circ}[/tex]
Therefore, the measure of angle A is 67.38°, and the expression used to find the measure of angle A is:
[tex]\cos^{-1}\left(\dfrac{5}{13}\right)[/tex]
