Trigonometric Ratios Worksheet for Calculating Sine, Cosine, and Tangent.

To find the trigonometric ratios (Sine, Cosine, and Tangent) for the mentioned angles, we use the definitions based on a right-angled triangle:

* Sine ($\sin$) = $\frac{\text{Opposite}}{\text{Hypotenuse}}$
* Cosine ($\cos$) = $\frac{\text{Adjacent}}{\text{Hypotenuse}}$
* Tangent ($\tan$) = $\frac{\text{Opposite}}{\text{Adjacent}}$

Key Rule: The Hypotenuse is always the longest side, opposite the right angle ($90^\circ$). The "Opposite" and "Adjacent" sides depend on which angle you are looking at.

Here is the step-by-step solution for each problem:

1) Angle $C$

* Identify Sides relative to $C$:
* Opposite side ($AB$) = $6$ ft
* Adjacent side ($BC$) = $8$ ft
* Hypotenuse ($AC$, opposite the right angle) = $10$ ft
* Calculate Ratios:
* $\sin C = \frac{6}{10}$ which simplifies to $\frac{3}{5}$ (or $0.6$)
* $\cos C = \frac{8}{10}$ which simplifies to $\frac{4}{5}$ (or $0.8$)
* $\tan C = \frac{6}{8}$ which simplifies to $\frac{3}{4}$ (or $0.75$)

2) Angle $F$

* Identify Sides relative to $F$:
* Opposite side ($EG$) = $8$ m
* Adjacent side ($EF$) = $11$ m
* Hypotenuse ($FG$, opposite the right angle) = $14$ m
* Calculate Ratios:
* $\sin F = \frac{8}{14}$ which simplifies to $\frac{4}{7}$
* $\cos F = \frac{11}{14}$
* $\tan F = \frac{8}{11}$

3) Angle $X$

* Identify Sides relative to $X$:
* Opposite side ($YZ$) = $5\sqrt{3}$ in
* Adjacent side ($XY$) = $5$ in
* Hypotenuse ($XZ$, opposite the right angle) = $10$ in
* Calculate Ratios:
* $\sin X = \frac{5\sqrt{3}}{10}$ which simplifies to $\frac{\sqrt{3}}{2}$
* $\cos X = \frac{5}{10}$ which simplifies to $\frac{1}{2}$
* $\tan X = \frac{5\sqrt{3}}{5}$ which simplifies to $\sqrt{3}$

4) Angle $B$

* Identify Sides relative to $B$:
* Opposite side ($AC$) = $4$ yd
* Adjacent side ($BC$) = $7$ yd
* Hypotenuse ($AB$, opposite the right angle) = $\sqrt{65}$ yd
* Calculate Ratios:
* $\sin B = \frac{4}{\sqrt{65}}$
* *(Optional: Rationalize denominator $\rightarrow \frac{4\sqrt{65}}{65}$)*
* $\cos B = \frac{7}{\sqrt{65}}$
* *(Optional: Rationalize denominator $\rightarrow \frac{7\sqrt{65}}{65}$)*
* $\tan B = \frac{4}{7}$

Final Answer:
1) Angle C:
$\sin C = \frac{3}{5}, \cos C = \frac{4}{5}, \tan C = \frac{3}{4}$

2) Angle F:
$\sin F = \frac{4}{7}, \cos F = \frac{11}{14}, \tan F = \frac{8}{11}$

3) Angle X:
$\sin X = \frac{\sqrt{3}}{2}, \cos X = \frac{1}{2}, \tan X = \sqrt{3}$

4) Angle B:
$\sin B = \frac{4}{\sqrt{65}}, \cos B = \frac{7}{\sqrt{65}}, \tan B = \frac{4}{7}$