Naming the Sides of a Right-Angled Triangle

The side opposite the right angle is called the hypotenuse.  It is the largest side of a right-angled triangle. If you stand at A in the triangle ABC, the side BC is opposite to you and the side AB is next to you.  We therefore say that BC is the opposite side to angleA and AB is the adjacent side to angle A. Notation Trigonometric Ratios
If you have trouble remembering the definitions, just remember SOH CAH TOA. Recall that: This is abbreviated as follows: Finding Side-Lengths We can use the trigonometric ratios sine, cosine and tangent to find any side-length of a right-angled triangle if we know another angle and one side-length. Using the Sine Ratio Example 1 Find the value of the pronumeral, correct to 2 decimal places, in the given diagram. Solution: Example 2 Find the length of the hypotenuse, correct to 2 decimal places, in the given diagram.          Solution: Let the length of the hypotenuse be x cm. So, the length of the hypotenuse is 55.01 cm. Using the Cosine Ratio Example 3 Find the value of the pronumeral in the following diagram, correct to 2 decimal places. Solution: Example 4 Find the length of the hypotenuse in the following diagram, correct to 2 decimal places. Solution: Let the length of the hypotenuse be x cm. So, the length of the hypotenuse is 109.45 cm. Using the Tangent Ratio Example 5 Find the value of the pronumeral in the following diagram, correct to 2 decimal places. Solution: Example 6 Find the value of the pronumeral in the following diagram, correct to 2 decimal places. Solution: Finding Angles We can use inverse trigonometric functions to find the angle(s) of a right-angled triangle if we know the length of any two sides. Using Inverse Sine Example 7 Solution: Example 8 Solution: Using Inverse Cosine Example 9 Solution: Using Inverse Tangent Example 10 Solution: Composite Figures A diagram consisting of more than one triangle is said to be a composite figure.For trigonometric problems involving a composite figure, first decide whether to use sine, cosine or tangent, and then calculate the required length or angle. Example 11 In the given diagram, find: a.  xb.  y Solution: Using a Construction Line To solve some trigonometric problems, we need to convert the given triangle into two right-angled triangles by drawing a perpendicular construction line from the vertex to the opposite side. Example 12 Find BC in the given diagram, rounded to 2 decimal places. Solution: Draw BD perpendicular to AC.  Let BD = x cm, BC = y cm.