The Cosine Rule
This page delves into the Cosine Rule, which is used in two specific scenarios: when given two sides and their included angle, or when all three sides of a triangle are known.
Cosine rule formula for side: a² = b² + c² - 2bc cos A
Cosine rule formula for angle: cos A = b2+c2−a2 / 2ac
The page provides two key examples:
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Calculating side length with cosine rule example:
Find the length of side BC in a triangle where two sides are 7cm and 3cm, with an included angle of 35°.
Example: Using the Cosine Rule:
a² = 7² + 3² - 273cos35°
a² = 49 + 9 - 34.40
a² = 23.60
a = √23.60 ≈ 4.9cm
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Finding angle using Cosine Rule calculator:
Find the size of angle B in a triangle with sides 4cm, 5cm, and 7cm.
Example: Applying the Cosine Rule for angles:
cos B = 42+52−72 / 2×4×5
cos B = 16+25−49 / 40
cos B = -0.2
B = cos⁻¹−0.2 ≈ 101.5°
These examples illustrate how the Cosine Rule can be used to find both sides and angles in triangles, making it a versatile tool in trigonometry. The page emphasizes the importance of choosing the correct formula based on the given information and what needs to be calculated.
Highlight: When using the Cosine Rule, pay attention to whether you're solving for a side or an angle, as the formulas differ slightly.
Understanding when and how to apply the Cosine Rule is crucial for students tackling complex triangle problems in trigonometry and geometry.