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Sine Rule

Geometry and Measures Sine Rule

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What is the Sine Rule Formula? (0:01)

The Sine Rule helps find missing sides or angles in non-right-angled triangles.

Sine Rule Formula:

For any triangle with angles A, B, C and opposite sides a, b, c, the Sine Rule states:

$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} $$$

or rearranged as:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $$$

When to Use the Sine Rule?

To find a missing side when you know one side and two angles.
To find a missing angle when you know two sides and one angle.

How to Use Sine Rule to Find Missing Sides? (0:30)

The Sine Rule is useful for finding missing sides in non-right-angled triangles when you know one angle and its opposite side.

Identify Given Information
- You know one angle (30°) and its opposite side (4 cm).
- You also know another angle (70°) and need to find its opposite side.
Apply the Sine Rule
- Using the Sine Rule formula: $$ \frac{\sin A}{a} = \frac{\sin B}{b} $$
- Substitute the known values: $$ \frac{\sin(30^\circ)}{4 \text{ cm}} = \frac{\sin(70^\circ)}{?} $$
Solve for the Missing Side
1. Rearranging the equation: $$
  ?= \frac{4 \text{ cm} \times \sin(70^\circ)}{\sin(30^\circ)}
  $$
2. Now, calculate the value to find the missing side. $$
  ? \approx 7.52 \text{ cm} \quad (\text{rounded to 2 decimal places}) $$

Exam Tip: When using the Sine Rule, always set up the equation carefully by matching each angle to its opposite side.

How to Use Sine Rule to Find Missing Angles? (1:25)

Identify Given Information
- You know one angle $135^\circ$ and its opposite side $10 \text{ cm}$.
- You also know another side $5 \text{ cm}$ and need to find its opposite angle $\theta$.
Apply the Sine Rule
- Using the Sine Rule formula:
  $$ \frac{\sin A}{a} = \frac{\sin B}{b} $$
- Substituting the known values: $$ \frac{\sin(135^\circ)}{10 \text{ cm}} = \frac{\sin(\theta)}{5 \text{ cm}} $$
Solve for the Missing Angle
- Rearranging the equation:
  $$ \sin(\theta) = \frac{5 \text{ cm} \times \sin(135^\circ)}{10 \text{ cm}} $$
- Now, since $ \sin(135^\circ) \approx 0.71 $, $$
  \sin(\theta) \approx \frac{5 \times 0.71}{10} \approx 0.355
  $$
- Now, use the inverse sine function to find the missing angle. $$ \theta \approx \sin^{-1} (0.355) $$
- On your calculator, press sin⁻¹, then enter 0.3555, and press = to get the angle. $$ \theta \approx 20.78^\circ \quad \text{(rounded to 2 decimal places)} $$

Exam Tip: When using the Sine Rule to find angles, always apply $\sin^{-1}$ (inverse sine) to get the angle.

📂 Revision Cards

Sine Rule Formulas for finding missing sides or angles. — 1) Sine Rule

Sine Rule applied to a triangle with angles 30°, 70°, and one unknown angle, showing how to find the opposite side of 4 cm using the sine rule. — 2) Applying the Sine Rule to find Missing Sides

Applying the sine rule to find angles with a triangle, with the equation sin(135°)/10 cm = sin(?)/5 cm. — 3) Applying the Sine Rule to find Missing Angles

🍪 Quiz Time - Practice Now!

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Sine Rule

1 / 6

Q: What information is required to apply the Sine Rule?

Three angles of the triangle

One pair of an angle and its opposite side

Three sides of the triangle

Feedback: Not quite! To apply the Sine Rule, you need to know at least one pair of an angle and its opposite side in the triangle.

Feedback: Well done! To apply the Sine Rule, you need to know at least one pair of an angle and its opposite side in the triangle.

2 / 6

Q: In the Sine Rule formula $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$, what do the letters a, b, and c represent?

The sides adjacent to the angles

The sides opposite to the angles

The area of the triangle

The angles of the triangle

Feedback: Not quite! In the Sine Rule formula, $a$, $b$, and $c$ represent the sides of the triangle, and A, B, and C are the angles opposite those sides.

Feedback: Well done! In the Sine Rule formula, $a$, $b$, and $c$ represent the sides of the triangle, and A, B, and C are the angles opposite those sides.

3 / 6

Q: In a triangle, angle A = 30°, and side a = 5 cm. If angle B = 70°, what is the length of side b?

8.5 cm

10.6 cm

9.4 cm

7.4 cm

Feedback: Not quite! Using the Sine Rule, $\frac{a}{\sin(A)} = \frac{b}{\sin(B)}$. Substituting the known values, $\frac{5}{\sin(30^\circ)} = \frac{b}{\sin(70^\circ)}$. Since $\sin(30^\circ) = 0.5$ and $\sin(70^\circ) \approx 0.94$, we cross-multiply to get $b \approx \frac{5 \times 0.94}{0.5} \approx 9.4$ cm.

Feedback: Well done! Using the Sine Rule, $\frac{a}{\sin(A)} = \frac{b}{\sin(B)}$. Substituting the known values, $\frac{5}{\sin(30^\circ)} = \frac{b}{\sin(70^\circ)}$. Since $\sin(30^\circ) = 0.5$ and $\sin(70^\circ) \approx 0.94$, we cross-multiply to get $b \approx \frac{5 \times 0.94}{0.5} \approx 9.4$ cm.

4 / 6

Q: In a triangle, angle A = 45°, and side a = 6 cm. If angle B = 60°, what is the length of side b?

6.9 cm

9.1 cm

7.7 cm

8.3 cm

Feedback: Not quite! Using the Sine Rule, $\frac{a}{\sin(A)} = \frac{b}{\sin(B)}$. Substituting the known values, $\frac{6}{\sin(45^\circ)} = \frac{b}{\sin(60^\circ)}$. Since $\sin(45^\circ) \approx 0.7$ and $\sin(60^\circ) \approx 0.9$, we cross-multiply to get $b \approx \frac{6 \times 0.9}{0.7} \approx 7.7$ cm.

Feedback: Well done! Using the Sine Rule, $\frac{a}{\sin(A)} = \frac{b}{\sin(B)}$. Substituting the known values, $\frac{6}{\sin(45^\circ)} = \frac{b}{\sin(60^\circ)}$. Since $\sin(45^\circ) \approx 0.7$ and $\sin(60^\circ) \approx 0.9$, we cross-multiply to get $b \approx \frac{6 \times 0.9}{0.7} \approx 7.7$ cm.

5 / 6

Q: In a triangle, angle A = 30°, and side a = 4 cm. If angle B = 100°, what is the length of side b?

8.0 cm

7.2cm

10.4 cm

9.6 cm

Feedback: Not quite! Using the Sine Rule, $\frac{a}{\sin(A)} = \frac{b}{\sin(B)}$. Substituting the known values, $\frac{4}{\sin(30^\circ)} = \frac{b}{\sin(100^\circ)}$. Since $\sin(30^\circ) = 0.5$ and $\sin(100^\circ) \approx 1.0$, we cross-multiply to get $b \approx \frac{4 \times 1.0}{0.5} \approx 8.0$ cm.

Feedback: Well done! Using the Sine Rule, $\frac{a}{\sin(A)} = \frac{b}{\sin(B)}$. Substituting the known values, $\frac{4}{\sin(30^\circ)} = \frac{b}{\sin(100^\circ)}$. Since $\sin(30^\circ) = 0.5$ and $\sin(100^\circ) \approx 1.0$, we cross-multiply to get $b \approx \frac{4 \times 1.0}{0.5} \approx 8.0$ cm.

6 / 6

Q: In a triangle, side a = 5 cm, side b = 10 cm, angle B = 120°, find angle A.

28.4°

25.8°

32.4°

36.2°

Feedback: Not quite! Using the Sine Rule, $\frac{a}{\sin(A)} = \frac{b}{\sin(B)}$. Substituting the known values, $\frac{5}{\sin(A)} = \frac{10}{\sin(120^\circ)}$. Since $\sin(120^\circ) \approx 0.866$, we can rearrange the equation to get $\sin(A) \approx \frac{5 \times 0.866}{10} \approx 0.433$. Finally, using the inverse sine function, we find that angle $A \approx 25.8^\circ$.

Feedback: Well done! Using the Sine Rule, $\frac{a}{\sin(A)} = \frac{b}{\sin(B)}$. Substituting the known values, $\frac{5}{\sin(A)} = \frac{10}{\sin(120^\circ)}$. Since $\sin(120^\circ) \approx 0.866$, we can rearrange the equation to get $\sin(A) \approx \frac{5 \times 0.866}{10} \approx 0.433$. Finally, using the inverse sine function, we find that angle $A \approx 25.8^\circ$.

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