OK Sine Cosine rules please

Question

ScanSolve AI Tutor · Accepted Answer

Okay, let's cover the Sine and Cosine Rules. These are used for solving non-right-angled triangles.

The Sine Rule relates the sides of a triangle to the sines of its opposite angles. It's used when you have:

For a triangle with sides a, b, c and opposite angles A, B, C respectively:

a/(sin A) = b/(sin B) = c/(sin C)

Or, to find an angle:

(sin A)/a = (sin B)/b = (sin C)/c

Example:
In a triangle, if side a = 10 cm, angle A = 30°, and angle B = 45°, find side b.

Write the Sine Rule formula.
a/(sin A) = b/(sin B)
Substitute the given values.
10/(sin 30°) = b/(sin 45°)
Solve for b.
b = 10 × (sin 45°)/(sin 30°)
b = 10 × (0.7071)/(0.5)
b = 10 × 1.4142
b = 14.14 cm (approx.)

Cosine Rule

The Cosine Rule relates the sides of a triangle to the cosine of one of its angles. It's used when you have:
1. Two sides and the included angle (SAS).
2. All three sides (SSS).
For a triangle with sides a, b, c and opposite angles A, B, C respectively:

To find a side:
a² = b² + c² - 2bc cos A
b² = a² + c² - 2ac cos B
c² = a² + b² - 2ab cos C

To find an angle:
cos A = (b² + c² - a²)/(2bc)
cos B = (a² + c² - b²)/(2ac)
cos C = (a² + b² - c²)/(2ab)

Example:
In a triangle, if side b = 8 cm, side c = 5 cm, and angle A = 60°, find side a.
Write the Cosine Rule formula to find side a.
a² = b² + c² - 2bc cos A
Substitute the given values.
a² = 8² + 5² - 2 × 8 × 5 × cos 60°
Calculate the squares and product.
a² = 64 + 25 - 2 × 8 × 5 × 0.5
a² = 89 - 80 × 0.5
a² = 89 - 40
a² = 49
Take the square root to find a.
a = √49
a = 7 cm

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