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# Opposite Side|Definition & Meaning

## Definition

A **triangle’s **opposite side is the side that lies **opposite **to the angle of interest and is not one of the **sides** that make the angle. An opposite side is always associated with **reference** to an angle. Without the **angle** reference, it is just a side of a triangle.

**Figure 1** shows the **opposite** side **PR** to the **angle s** in a triangle PQR.

## Right-angle Triangle

A right-angle **triangle** is characterized by a** right**(**90°**) angle. Its two sides are **perpendicular** to each other. **Trigonometry** is based on the relationship between the **angles** and **sides** of a right-angle triangle.

A right-angle triangle consists of a **hypotenuse** which is the **opposite** side to the **right angle** and is the **longest** side. In this case, the angle of interest is the **90°** angle.

In **contrast** to the opposite side, the **adjacent** side makes the **angle** of interest. If an angle theta(**θ**) is formed between the **hypotenuse** and the **adjacent** side, then the **perpendicular** side will be opposite to **θ**.

Sometimes, the **adjacent** side is referred to as the **base**, and the **opposite** side is the **perpendicular** in the right-angle triangle.

**Figure 2** shows a right-angle triangle **ABC** with **AC** as the hypotenuse, the adjacent side as **BC**, and the opposite side as **AB** with reference to angle** θ**.

### Pythagoras Theorem

The Pythagoras theorem is only for **right-angle** triangles. It states that **hypotenuse** square is equal to the sum of the **square** of the **base **b(**adjacent** side) length and the **perpendicular** p(**opposite** side) length.

Mathematically, it is:

h^{2} = p^{2} + $b^{2}

**Figure 3** shows the **right-angle** triangle with hypotenuse **h**, perpendicular **p**, and the base **b**.

Where **LN** is the **opposite** side, and **MN** is the **adjacent** side of angle **θ**.

### Trigonometric Functions

Trigonometric **functions** are the relationship between an **angle** and the ratio of two **side** lengths of a **right-angle** triangle. These are sine, cosine, and tangent, which are mostly used. The **notation** for sine is “**sin**,” cosine is “**cos,**” and for tangent is “**tan**.”

Consider **figure 2** for the trigonometric **ratios**. If the length of the side “**a”** is **opposite** to angle **θ**, **b** is the **adjacent** side, and **c** is the **hypotenuse**, then:

**sin θ** = opposite / hypotenuse = a / c

**cos θ** = adjacent / hypotenuse = b / c

**tan θ** = opposite / adjacent = a / b

The angle **θ** should be **acute** i.e. between **0°** and **90°**. These ratios suggest that the ratio of any two **sides** of the triangle **depends** on the angle **θ**.

The three **ratios** are related as follows:

tan θ = sin θ / cos θ

The three **angles** of a triangle add up to **180°. **Hence, a right-angle triangle has two **acute** angles that sum up to **90°**.

### Less Common Trigonometric Functions

The **reciprocals** of the above trigonometric **functions** are the less-used functions. These are cosecant, **secant**, and cotangent. The notations for **cosecant**, secant, and **cotangent** are cosec or csc, sec, and cot, respectively. These **ratios** are given as follows:

**cosec θ** = 1 / sin θ = hypotenuse / opposite = c / a

**sec θ** = 1 / cos θ = hypotenuse / adjacent = c / b

**cot θ** = 1 / tan θ = adjacent / opposite = b / a

### Semi-perimeter

Semi-perimeter is defined as **half** of the **perimeter** of a polygon. The following **formula** gives the semi-perimeter **s** of the three **sides ****a**, **b**, and **c **of the triangle as follows:

s = (a + b + c) / 2

If **a** is the **opposite** side, **b** is the **adjacent** side and **c** is the **hypotenuse** of a right-angle triangle, then the equation for the **semi-perimeter** is given as:

(s – a)(s – b) = s(s – c)

## Law of Cosine

The law of cosine deals with the **opposite** side of the triangle. It states that the **square** of a side **opposite** to an angle is equal to the **sum** of the square of the remaining two **sides** that make the angle minus two times the **product** of the remaining sides and the **cosine** of that angle.

Consider **A**, **B**, and **C** as three **angles** of a triangle, as shown in **figure 4**. Let the sides **b**, **c,** and **a** be **opposite** to the respective angles B, C, and A.

The three **equations** for the **law **of** cosine** are given as follows:

b^{2} = c^{2} + a^{2} – 2ca.cos(B)

c^{2} = b^{2} + a^{2} – 2ba.cos(C)

a^{2} = c^{2} + b^{2} – 2cb.cos(A)

The law of cosine is **helpful** in finding the **third** side of a **triangle** when its **opposite** angle and the **sides** enclosing it are known. It can also be used to find the **angle** opposite to a side if the **three** sides are known.

The cosine rule **generalizes** the Pythagoras theorem, which is only for **right** angles. If angle **A = 90°**, then** cos A = 0**, hence the first equation for the** cosine rule** becomes:

a^{2} = b^{2} + c^{2}

which is the **Pythagoras** theorem.

## Examples

### Example 1 – Identifying Opposite Sides

Three **triangles** are shown in **figure 5**. In each triangle, identify the **opposite** side and the **adjacent** side to the angle **θ**. Which of the three triangles is a **right-angle** triangle? Also, identify the **hypotenuse** and the right angle in the right-angle triangle.

### Solution

In triangle (a) **PQR**, the **opposite** side to angle θ is **PR,** and the two **adjacent** sides to angle θ are **PQ** and **QR,** which constitutes the angle.

In triangle (b)** LMN**, the **opposite** side to angle θ is **LM,** and the **adjacent** sides are **LN** and **MN**.

The third triangle (c) **HIJ** is a **right-angle** triangle with a **90°** angle HIJ. The side **opposite** to angle θ is **JI,** and the **adjacent** side is **IH**. The **hypotenuse** is **HJ,** as it is the longest side.

### Example 2 – Calculating the Length of the Opposite Side Using the Trigonometric Functions

A **triangle** MNO is given in **figure 6**. Calculate the length of the side **opposite** to angle **θ**. The **hypotenuse** is given as **4.9 cm**.

### Solution

The **opposite** side to angle **θ** is **OM**. Using the function:

sin θ = opposite / hypotenuse

This **equation** can be written as:

Opposite = (sin θ)(hypotenuse)

Here, θ = **35°** and hypotenuse = **4.9 cm**, putting the values in the above equation gives:

Opposite = (sin 35°)(4.9)

Opposite = (0.57)(4.9)

**Opposite = 2.8 cm**

So, the opposite side **OM** is equal to **2.8 cm**.

### Example 3 – Calculating the Opposite Side Length Using the Law of Cosine

**Figure 7** shows a triangle. Calculate the side **opposite** to angle **A**.

### Solution

The **opposite** side “**a**” to angle **A** can be calculated using the **cosine** rule:

a^{2} = c^{2} + b^{2} – 2cb.cos(A)

Where:

b = 10 m, c = 7 m, angle A = 38°

Putting the **values** in the above equation gives:

a^{2} = (7)^{2} + (10)^{2} – 2(7)(10)cos(38°)

a^{2} = 49 + 100 – 140(0.788)

a^{2} = 149 – 110.32

a^{2} = 38.68

Taking **square root** on both sides gives:

**a = 6.22 m**

So, the **side opposite** to angle** A** is equal to **6.22 m**.

*All the images are created using GeoGebra.*