Question

here is triangle abc.
select all the true equations.
a (sin(27)=\frac{y}{15})
b (cos(63)=\frac{y}{15})
c (\tan(27)=\frac{y}{x})
d (sin(63)=\frac{x}{15})
e (\tan(63)=\frac{y}{x})

Explanation:

First, recall the definitions of sine, cosine, and tangent in a right triangle. In right triangle \(ABC\) with right angle at \(C\), hypotenuse \(AB = 15\), opposite side to angle \(A\) (27°) is \(BC = y\), adjacent side to angle \(A\) is \(AC = x\), and opposite side to angle \(B\) (63°) is \(AC = x\), adjacent side to angle \(B\) is \(BC = y\).

Step 1: Analyze Option A

\(\sin(27^\circ)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{y}{15}\), not \(\frac{x}{15}\). So A is false.

Step 2: Analyze Option B

\(\cos(63^\circ)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{y}{15}\) (since angle at B is 63°, adjacent side is \(y\), hypotenuse 15). So B is true.

Step 3: Analyze Option C

\(\tan(27^\circ)=\frac{\text{opposite}}{\text{adjacent}}=\frac{y}{x}\). So C is true.

Step 4: Analyze Option D

\(\sin(63^\circ)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{x}{15}\) (opposite side to 63° is \(x\), hypotenuse 15). So D is true.

Step 5: Analyze Option E

\(\tan(63^\circ)=\frac{\text{opposite}}{\text{adjacent}}=\frac{x}{y}\), not \(\frac{y}{x}\). So E is false.

Answer:

B. \(\cos(63)=\frac{y}{15}\), C. \(\tan(27)=\frac{y}{x}\), D. \(\sin(63)=\frac{x}{15}\)