Question

directions: find the desired ratios of the given triangle. write your answers as simplified fractions. use the triangle above to fill in the blanks. write your answers as simplified fractions. a. cos(b)=□ b. sin(a)=□ c. sin(?)=1/2 d. ?(a)=2√3/4 = √3/2 e. tan(?)=2√3/2 = √3 hint: if your answer includes a square root, you must write your answer as 11sqrt(5) if it is supposed to look like 11√5 you have 3 attempts to correctly answer this problem. > next question

Explanation:

Step1: Recall cosine formula

$\cos(B)=\frac{\text{adjacent}}{\text{hypotenuse}}$. In right - triangle $ABC$, for angle $B$, the adjacent side to $B$ is $BC = 2\sqrt{3}$ and the hypotenuse $AB$ can be found using the Pythagorean theorem $AB=\sqrt{(2\sqrt{3})^2 + 2^2}=\sqrt{12 + 4}=\sqrt{16}=4$. So $\cos(B)=\frac{2\sqrt{3}}{4}=\frac{\sqrt{3}}{2}$.

Step2: Recall sine formula

$\sin(A)=\frac{\text{opposite}}{\text{hypotenuse}}$. For angle $A$, the opposite side is $BC = 2\sqrt{3}$ and the hypotenuse $AB = 4$. So $\sin(A)=\frac{2\sqrt{3}}{4}=\frac{\sqrt{3}}{2}$.

Step3: Recall sine values

We know that $\sin(30^{\circ})=\frac{1}{2}$. In a right - triangle, if $\sin(\theta)=\frac{1}{2}$, and considering the angles of a right - triangle, $\theta = 30^{\circ}$. In this triangle, angle $A = 60^{\circ}$ and angle $B = 30^{\circ}$, so the angle is $B$.

Step4: Recall cosine values

We have $\frac{2\sqrt{3}}{4}=\frac{\sqrt{3}}{2}$, and $\cos(A)=\frac{\text{adjacent}}{\text{hypotenuse}}$. For angle $A$, the adjacent side is $AC = 2$ and the hypotenuse $AB = 4$, so the missing function is $\cos(A)$.

Step5: Recall tangent values

We have $\frac{2\sqrt{3}}{2}=\sqrt{3}$, and $\tan(B)=\frac{\text{opposite}}{\text{adjacent}}$. For angle $B$, the opposite side is $AC = 2$ and the adjacent side is $BC = 2\sqrt{3}$, so the missing angle is $B$.

Answer:

a. $\frac{\sqrt{3}}{2}$
b. $\frac{\sqrt{3}}{2}$
c. $B$
d. $\cos$
e. $B$