Certainly, I'll solve for each unknown side in the triangles using trigonometric ratios. These are all right triangles, so we can apply sine, cosine, and tangent functions.

Solution for each triangle:

(a)

• Given: Angle = 22°, adjacent side = 25, unknown = opposite side $y$.
• Using the tangent function: $\tan(22^\circ) = \frac{y}{25}$
• Solving for $y$: $y = 25 \cdot \tan(22^\circ) \approx 25 \cdot 0.4040 \approx 10.1$

(b)

• Given: Angle = 11°, hypotenuse = 100, unknown = adjacent side $x$.
• Using the cosine function: $\cos(11^\circ) = \frac{x}{100}$
• Solving for $x$: $x = 100 \cdot \cos(11^\circ) \approx 100 \cdot 0.9816 \approx 98.16$

(c)

• Given: Angle = 45°, adjacent side = 13, unknown = opposite side $a$.
• Using the tangent function: $\tan(45^\circ) = \frac{a}{13}$
• Since $\tan(45^\circ) = 1$: $a = 13$

(d)

• Given: Right triangle with both legs equal to 4, unknown = hypotenuse $o$.
• Using the Pythagorean theorem: $o = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} \approx 5.66$

(e)

• Given: Opposite side = 20, adjacent side = 50, unknown angle $\theta$.
• Using the tangent function: $\tan(\theta) = \frac{20}{50} = 0.4$
• Solving for $\theta$: $\theta = \tan^{-1}(0.4) \approx 21.8^\circ$

(f)

• Given: Angle = 79°, adjacent side = 7, unknown = opposite side $b$.
• Using the tangent function: $\tan(79^\circ) = \frac{b}{7}$
• Solving for $b$: $b = 7 \cdot \tan(79^\circ) \approx 7 \cdot 5.1446 \approx 36.01$

Summary of Answers:

1. (a) $y \approx 10.1$
2. (b) $x \approx 98.16$
3. (c) $a = 13$
4. (d) $o \approx 5.66$
5. (e) $\theta \approx 21.8^\circ$
6. (f) $b \approx 36.01$

Would you like a more detailed breakdown of any specific triangle or the trigonometric functions used?

Related Questions:

1. What trigonometric ratios are best suited for finding missing sides in right triangles?
2. How do you determine which trigonometric function to use based on given sides?
3. Can you solve these problems without a calculator? What approximations would you need?
4. How would the calculations change if the triangles were not right triangles?
5. Why is the tangent of 45° equal to 1?

Tip:

Remember, in right triangles, the hypotenuse is always the longest side, and knowing this can help verify your calculations.