To solve the given problems, we'll apply trigonometric functions (sine, cosine, and tangent) as well as the Pythagorean theorem, which are the standard methods for solving right triangles.

Problem 6:

You have a right triangle where:

• The side opposite to the 66° angle is $5.5 \, m$ (this is side $a$).
• The hypotenuse is $h$.
• The adjacent side is $q$ (this is side $b$).

a) Finding the hypotenuse $h$:

We use the sine function since we know the opposite side and the angle: $\sin(66^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{h} = \frac{5.5}{h}$ Solving for $h$: $h = \frac{5.5}{\sin(66^\circ)}$ $h \approx \frac{5.5}{0.9135} \approx 6.0 \, m$

b) Finding side $b$ (adjacent side):

We use the cosine function for this: $\cos(66^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{b}{h}$ $b = h \cdot \cos(66^\circ) = 6.0 \cdot 0.4067 \approx 2.4 \, m$

Problem 7:

You have a right triangle where:

• The hypotenuse $h$ is $35.5 \, cm$.
• The angle is $70^\circ$.
• Side $b$ is opposite to the 70° angle.
• Side $c$ is adjacent to the 70° angle.

a) Finding side $b$ (opposite side):

We use the sine function here: $\sin(70^\circ) = \frac{b}{h} = \frac{b}{35.5}$ $b = 35.5 \cdot \sin(70^\circ) = 35.5 \cdot 0.9397 \approx 33.4 \, cm$

b) Finding side $c$ (adjacent side):

We use the cosine function: $\cos(70^\circ) = \frac{c}{h} = \frac{c}{35.5}$ $c = 35.5 \cdot \cos(70^\circ) = 35.5 \cdot 0.3420 \approx 12.1 \, cm$

Problem 8:

You have a right triangle where:

• The adjacent side $c$ is $100 \, m$.
• The angle is $27^\circ$.
• Side $a$ is opposite to the 27° angle.
• The hypotenuse is $h$.

a) Finding side $a$ (opposite side):

We use the tangent function here: $\tan(27^\circ) = \frac{a}{c} = \frac{a}{100}$ $a = 100 \cdot \tan(27^\circ) = 100 \cdot 0.5095 \approx 50.95 \, m$

b) Finding the hypotenuse $h$:

We use the cosine function: $\cos(27^\circ) = \frac{c}{h}$ $h = \frac{c}{\cos(27^\circ)} = \frac{100}{0.8910} \approx 112.3 \, m$

These are the solutions to the problems. Would you like more details or have any further questions?

5 Related Questions:

1. What is the general formula for solving right triangles?
2. How do you know when to use sine, cosine, or tangent?
3. Can you explain the Pythagorean theorem and how it relates to right triangles?
4. How do angle and side length changes affect the trigonometric ratios in a triangle?
5. Can you use trigonometry to solve non-right triangles?

Tip:

When solving for side lengths in right triangles, always make sure to choose the correct trigonometric function based on the known and unknown sides (opposite, adjacent, or hypotenuse).