Question

find the sine, cosine, and tangent of ∠b.

simplify your answers and write them as proper fractions, improper fractions, or whole numbers.

sin(b) =
cos(b) =
tan(b) =

Explanation:

Step1: Find length of BC

In right triangle \(BCD\), by Pythagoras theorem \(BD^{2}=BC^{2}+CD^{2}\). Given \(BD = 5\), \(CD = 4\), so \(BC=\sqrt{BD^{2}-CD^{2}}=\sqrt{5^{2}-4^{2}}=\sqrt{25 - 16}=\sqrt{9}=3\).

Step2: Calculate \(\sin(B)\)

\(\sin(B)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{CD}{BD}=\frac{4}{5}\).

Step3: Calculate \(\cos(B)\)

\(\cos(B)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{BC}{BD}=\frac{3}{5}\).

Step4: Calculate \(\tan(B)\)

\(\tan(B)=\frac{\text{opposite}}{\text{adjacent}}=\frac{CD}{BC}=\frac{4}{3}\).

Answer:

\(\sin(B)=\frac{4}{5}\), \(\cos(B)=\frac{3}{5}\), \(\tan(B)=\frac{4}{3}\)