In trigonometry, we can expand upon the principles of right-angled triangles and apply them to non-right angled triangles. Two essential rules for solving these types of triangles are known as the sine and cosine rules. These rules provide a way to find unknown angles and lengths in non-right angled triangles.

When labelling a triangle, it is important to use lowercase letters for the sides and uppercase letters for the angles. The same letter is assigned to the opposite sides and angles. For example, in the triangle below, we label **A** opposite **a**, **B** opposite **b**, and **C** opposite **c**. **INSERT IMAGE OF LABELLED TRIANGLE HERE**

For a triangle with the above labelling, the sine rule formula can be written as:

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

Or, in another form:

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

The sine rule is commonly used in two scenarios:

- When two angles and one side are known, and we need to find the length of another side.
- When two lengths and one angle are known, and we need to find another angle.

For example:

**Example 1:** Find **x**.

Using the sine rule, we know that a/sin(A) = b/sin(B). Rearranging for **x**, we get bsin(A)/sin(B). By plugging in the known values, we get x = bsin(40°)/sin(70°) = 23.6 (3 sig fig).

**Example 2:** Find **y**.

Using the sine rule, we know that sin(A)/a = sin(C)/c. By rearranging, we get asin(C)/c. By substituting the values, we get y = (30)(sin(40°))/(15) = 54.5° (3 sig fig).

The cosine rule is utilized for non-right angled triangles when we know three lengths and one angle. The formula for the cosine rule is given by:

a² = b² + c² - 2bc · cos(A)

To solve for an angle using the cosine rule, we rearrange the formula to get:

cos(A) = (b² + c² - a²)/2bc

For instance:

**Example 1:** Find **x**.

Using the cosine rule, we get x² = 15² + 19² - 2(15)(19)cos(40°) = 149.354667… By rearranging, we get x = √149.354667… = 12.2 (3 sig fig)

**Example 2:** Find **y**.

Using the rearranged cosine rule, we get y = (15² + 19² - 12²)/2(15)(19) = 27.7 (3 sig fig)

Now that we have learned how to use the sine and cosine rules, let's take a look at how they are derived from first principles. This requires some understanding of trigonometry and Pythagoras' theorem.

Beginning with the labelled triangle as described above, we can draw a line from the top angle to the base, creating two right-angled triangles. We will call this line **h**.

**INSERT IMAGE OF DERIVATION OF SINE RULE HERE**

In a similar approach, we can use the same triangle and draw a line from the top angle to the base, creating two right-angled triangles. This line will also be labelled **h**. We can also label the point where **h** meets the base as **D** and the two sides as **x** and **y**.

**INSERT IMAGE OF DERIVATION OF COSINE RULE HERE**

By applying Pythagoras' theorem to the left triangle, we get x² + h² = c². By rearranging, we get x = √(c² - h²). By plugging this value into the right triangle, we get y = √(a² - x²). By substituting these values into the formula for the cosine rule, we get:

a² = x² + y² - 2xy · cos(A)

= (c² - h²) + (a² - x²) - 2(c - h)(a - x) · cos(A)

= c² - h² + a² - x² + 2cx - 2ch · cos(A)

By using the substitution h = bcos(A), we get:

= c² - b²cos²(A) + a² - x² + 2bxcos(A) - 2bx · cos(A)

= c² - b²cos²(A) + a² - x² + 2bxcos(A) - 2bx · cos(A)

Simplifying further, we get a² = b² + c² - 2bc · cos(A).

Remember these key points when using the sine and cosine rules:

- The sine rule is utilized when we have one unknown value and three known values from two angles and two sides.
- The cosine rule is utilized when we have one unknown value and three known values from one angle and three sides.
- Both the sine and cosine rules can be derived from first principles.

In summary, the sine and cosine rules are essential tools for solving sides and angles in non-right angled triangles.

Trigonometry is a branch of mathematics that deals with the relationship between sides and angles in a triangle. Two important concepts in trigonometry are the sine and cosine rules, which are used for solving unknown values in triangles.

Both the sine and cosine rules are based on the definitions of sine and cosine in right angled triangles. The sine of an angle is found by dividing the length of the side opposite the angle by the length of the hypotenuse. Similarly, the cosine of an angle is found by dividing the length of the side adjacent to the angle by the length of the hypotenuse. These definitions are the foundation for the application of sine and cosine rules in non-right angled triangles.

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