The image you uploaded contains multiple right-angled triangle problems where the goal is to find unknown sides using trigonometric functions (sine, cosine, or tangent). Let’s work through a few of these problems step by step, using the provided angles and sides.

Formulae to use for right-angled triangles:

• Sine rule: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$
• Cosine rule: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$
• Tangent rule: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

Let's solve for each one:

Problem 1:

• Angle: $27^\circ$
• Adjacent side: 10 cm
• Unknown side: $a$ (hypotenuse)

Using the cosine rule: $\cos(27^\circ) = \frac{10}{a} \quad \Rightarrow \quad a = \frac{10}{\cos(27^\circ)}$

Problem 2:

• Angle: $61^\circ$
• Adjacent side: 4 cm
• Unknown side: $b$ (hypotenuse)

Using the cosine rule: $\cos(61^\circ) = \frac{4}{b} \quad \Rightarrow \quad b = \frac{4}{\cos(61^\circ)}$

Problem 3:

• Angle: $58^\circ$
• Opposite side: 7 cm
• Unknown side: $c$ (hypotenuse)

Using the sine rule: $\sin(58^\circ) = \frac{7}{c} \quad \Rightarrow \quad c = \frac{7}{\sin(58^\circ)}$

Problem 4:

• Angle: $42.6^\circ$
• Opposite side: 9 cm
• Unknown side: $x$ (hypotenuse)

Using the sine rule: $\sin(42.6^\circ) = \frac{9}{x} \quad \Rightarrow \quad x = \frac{9}{\sin(42.6^\circ)}$

Problem 5:

• Angle: $38.7^\circ$
• Adjacent side: 10 cm
• Unknown side: $x$ (hypotenuse)

Using the cosine rule: $\cos(38.7^\circ) = \frac{10}{x} \quad \Rightarrow \quad x = \frac{10}{\cos(38.7^\circ)}$

Would you like me to calculate these values for you?

Related questions:

1. How do you decide which trigonometric rule to use?
2. What is the difference between the sine, cosine, and tangent functions?
3. How do you round trigonometric results to three significant figures?
4. What do you do when you’re given two sides but no angles?
5. Can you explain the use of inverse trigonometric functions?

Tip:

Remember, for a right-angled triangle, always identify the sides as opposite, adjacent, or hypotenuse based on the angle given. This will help you choose the correct trigonometric function to use.