Question

44 which one(s) of the following trig ratios is equal to \\(\frac{24}{25}\\)? select all that apply: triangle with right angle at a, sides: ac=7, at=24, ct=25 a \\(\sin(a)\\) b \\(\sin(t)\\) c \\(\sin(c)\\) d \\(\cos(a)\\) e \\(\cos(t)\\) f \\(\cos(c)\\) g \\(\tan(a)\\) h \\(\tan(t)\\) i \\(\tan(c)\\)

Explanation:

Step1: Recall Trig Ratios

In a right triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Triangle \(CAT\) is right - angled at \(A\), so \(AC = 7\), \(AT=24\), \(CT = 25\) (hypotenuse).

Step2: Analyze \(\sin(A)\)

For \(\angle A = 90^{\circ}\), \(\sin(90^{\circ})=1
eq\frac{24}{25}\).

Step3: Analyze \(\sin(T)\)

For \(\angle T\), opposite side to \(\angle T\) is \(AC = 7\), hypotenuse \(CT = 25\). So \(\sin(T)=\frac{AC}{CT}=\frac{7}{25}
eq\frac{24}{25}\).

Step4: Analyze \(\sin(C)\)

For \(\angle C\), opposite side to \(\angle C\) is \(AT = 24\), hypotenuse \(CT = 25\). So \(\sin(C)=\frac{AT}{CT}=\frac{24}{25}\).

Step5: Analyze \(\cos(A)\)

For \(\angle A=90^{\circ}\), \(\cos(90^{\circ}) = 0
eq\frac{24}{25}\).

Step6: Analyze \(\cos(T)\)

For \(\angle T\), adjacent side to \(\angle T\) is \(AT = 24\), hypotenuse \(CT = 25\). So \(\cos(T)=\frac{AT}{CT}=\frac{24}{25}\).

Step7: Analyze \(\cos(C)\)

For \(\angle C\), adjacent side to \(\angle C\) is \(AC = 7\), hypotenuse \(CT = 25\). So \(\cos(C)=\frac{AC}{CT}=\frac{7}{25}
eq\frac{24}{25}\).

Step8: Analyze \(\tan(A)\)

For \(\angle A = 90^{\circ}\), \(\tan(90^{\circ})\) is undefined.

Step9: Analyze \(\tan(T)\)

For \(\angle T\), opposite side \(AC = 7\), adjacent side \(AT = 24\). So \(\tan(T)=\frac{AC}{AT}=\frac{7}{24}
eq\frac{24}{25}\).

Step10: Analyze \(\tan(C)\)

For \(\angle C\), opposite side \(AT = 24\), adjacent side \(AC = 7\). So \(\tan(C)=\frac{AT}{AC}=\frac{24}{7}
eq\frac{24}{25}\).

Answer:

C. \(\sin(C)\), E. \(\cos(T)\)