Trigonometry is the study of the relationships between the angles and sides of triangles. It is a useful branch of mathematics that has many applications in science, engineering, navigation, surveying, and more. In this blog post, we will learn about the basic concepts of trigonometry, such as trigonometric ratios, and how to use them to solve problems involving right-angled triangles.
What are Trigonometric Ratios?
Trigonometric ratios are the ratios of the lengths of two sides of a right-angled triangle. A right-angled triangle is a triangle that has one angle of 90 degrees, and two acute angles (less than 90 degrees). The three sides of a right-angled triangle have special names:
The hypotenuse is the longest side of the triangle, and it is opposite to the right angle.
The adjacent side is the side that is next to (or adjacent to) the angle we are interested in.
The opposite side is the side that is opposite to the angle we are interested in.
There are six trigonometric ratios that relate the angle and the sides of a right-angled triangle. They are:
Sine (sin): The ratio of the opposite side to the hypotenuse.
Cosine (cos): The ratio of the adjacent side to the hypotenuse.
Tangent (tan): The ratio of the opposite side to the adjacent side.
Cosecant (cosec or csc): The reciprocal of sine, or the ratio of the hypotenuse to the opposite side.
Secant (sec): The reciprocal of cosine, or the ratio of the hypotenuse to the adjacent side.
Cotangent (cot): The reciprocal of tangent, or the ratio of the adjacent side to the opposite side.
We can write these ratios using symbols as follows:
where θ is the angle we are interested in.
How to Use Trigonometric Ratios?
Trigonometric ratios can help us find missing angles or sides in a right-angled triangle. For example, suppose we have a right-angled triangle with an angle of 30 degrees and a hypotenuse of 10 cm. We want to find the length of the opposite side.
We can use the sine ratio to find the opposite side, since we know the angle and the hypotenuse. We can write:
To find the opposite side, we can multiply both sides by 10:
Now we need to use a calculator to find the value of sin30∘. We can type in “30” and then press the “sin” button. We get:
So we can substitute this value into our equation:
And then simplify:
Therefore, the length of the opposite side is 5 cm.
We can use a similar process to find other missing angles or sides using different trigonometric ratios. The key is to identify which ratio involves the given information and the unknown quantity, and then use a calculator to find or use trigonometric values.
Trigonometric Ratios Table
It is useful to memorize some common values of trigonometric ratios for certain angles, such as 0, 30, 45, 60, and 90 degrees. These values can help us solve problems without using a calculator, or check our answers. The table below shows the values of the six trigonometric ratios for these angles:
Note that some values are undefined because they involve dividing by zero, which is not possible. Also note that the values of sine and cosine are symmetrical, meaning that sinθ=cos(90−θ) and cosθ=sin(90−θ). This is because the opposite and adjacent sides of a right-angled triangle are interchanged when we consider the complementary angle.
Trigonometric Ratios Mnemonics
Mnemonics are memory aids that help us remember things more easily. There are some mnemonics that can help us remember the definitions of the trigonometric ratios. For example, one common mnemonic is SOHCAHTOA, which stands for:
Sine = Opposite / Hypotenuse
Cosine = Adjacent / Hypotenuse
Tangent = Opposite / Adjacent
Another mnemonic is Some People Have Curly Brown Hair Through Proper Brushing, which stands for:
Sine = Perpendicular / Hypotenuse
Cosine = Base / Hypotenuse
Tangent = Perpendicular / Base
You can also make up your own mnemonics that suit your style and preference. The important thing is to understand the meaning and use of the trigonometric ratios, not just memorize them.
Solved Examples for You
Example 1: Find the value of tan 60 degrees using the trigonometric ratios table.
Solution: We can look up the value of tan 60 degrees in the table and see that it is equal to √3. Therefore, tan 60 degrees = √3.
Example 2: Find the length of the hypotenuse of a right-angled triangle with an angle of 45 degrees and an adjacent side of 8 cm.
Solution: We can use the cosine ratio to find the hypotenuse, since we know the angle and the adjacent side. We can write:
Substituting the given values, we get:
We can use the table to find the value of cos 45 degrees, which is 1/√2. Substituting this value, we get:
To find the hypotenuse, we can cross-multiply and simplify:
Therefore, the length of the hypotenuse is 8√2 cm.