Let's learn trigonometric ratio definition

Trigonometry is all about understanding the properties of right angled triangle. As we know that many complex polygons can be built by one or more triangles, trigonometry becomes important in that case. Even trigonometry ratios can be used to derive many universal facts. Say for example, you want to calculate the distance of moon from earth, then trigonometric ratios could be of great use.

Coming back to the topic, we have 6 trigonometry ratios in right triangles that are named sine, cos, tangent, cosecant, secant and cotangent. Consider the triangle given below. Having trigonometric ratio definition in mind, the values of all trigonometric ratios in right triangles are as given below.
trigonometric ratio

  • sinθ = opposite / hypotenuse

  • cosθ = adjacent / hypotenuse

  • tanθ = opposite / adjacent

  • cosecθ = hypotenuse / opposite

  • secθ = hypotenuse / adjacent

  • cotθ = adjacent / opposite

Lets study the table of trigonometric ratios followed by examples:

The trigonometric ratios table given below provides value of all the ratios for different standard values of θ as per the trigonometric ratios definition.

θ

00

300

450

600

900

sin θ

0

½

1/√2

√3/2

1

cos θ

1

√3/2

1/√2

½

0

tan θ

0

1/√3

1

√3

Not defined

cot θ

Not defined

√3

1

1/√3

0

sec θ

1

2/√3

√2

2

Not defined

cosec θ

Not defined

2

√2

2/√3

1

The trigonometric ratios examples given below will help to make this concept clear. Note that the values given in table of trigonometric ratios are of great use for solving this examples.

(1) Evaluate cos 30° - sin 60° + tan 45°:

From the table, we have cos 30° - sin 60° + tan 45° = √3/2 - √3/2 + 1 = 1.

(2) Evaluate cos2 45° + sin2 45°:

We have a trigonometric property cos2θ + sin2θ = 1 for any values of θ. Let us compare whether values from table yields same results or not. From the table, cos 45° = sin 45° = 1/√2. Putting this values in the main equation, cos2 45° + sin2 45° = ½ + ½ = 1. Thus the property is verified.

Study of inverse trigonometric ratios:

Suppose you want to find the angle provided that you have trigonometric values, inverse trigonometric ratios could be an solution. They are notified as sin-1 x, cos-1x, tan-1x and so on. Some times, they are also referred as arc cosec, arc sec etc. The output will be sin-1x = θ.

To find trigonometric ratios of allied angles, its mandatory to learn allied angles. Any two angles whose summation or difference comes in multiple of 90° are called allied angles. This concept makes the method of finding the ratio a bit easier. Now lets have a look at some of the trigonometric ratios word problems.

(1) Find θ if sinθ = 1/√2:

We have θ = sin-1 (1/√2) = 45°.

(2) Consider the above right triangle ABC with AB = 3, BC = 4 and CA = 5 units. Find sinθ, cosθ and tanθ:

We have sinθ = BC/CA = 4/5. cosθ = AB/CA = 3/5 and tanθ = BC/AB = 4/3. This shows that tanθ is equal to the ratio of sinθ and cosθ.

Thus we have seen the trig ratio word problems thereby having a detailed study of trigonometric ratios.

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