Trigonometry Angles

The word trigonometry means ‘study of triangles’. This study of triangles has been very important to the growth of human society. From architecture to astronomy there are very few fields that don't use trigonometry are very few. In this article, we are particularly concentrating on the properties of angles of the right triangle. A triangle with one angle equal to 900 is called a right-angled triangle/ right triangle. Triangles are classified into 3 groups based on the length of the sides of the triangle: The equilateral triangle, isosceles triangle, and Scalene triangle. The equilateral triangle is formed by three equal sides, the isosceles triangle has any two sides equal and the scalene triangle has all three sides of different lengths. 

Trigonometry Angle Measures

The length of the sides of triangles affects the angles formed by the triangles too. We  know,

  • The total angle inside a triangle should be 1800

  • The angles opposite to equal sides are equal.

These two principles applied to the triangles gives

  • All three angles of an equilateral triangle are equal to 600.

  • The two angles opposite to the two equal sides are equal.

  • All three angles of the scalene triangles are different.

Some Definitions

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Here a is the altitude, b is the base and ‘h’ is the hypotenuse. We can define some very useful ratios using sides and the angle θ. 

(θ) = \[\frac{\text{opposite}}{\text{hypotenuse}} side\]

\[\frac{a}{h} = Cosine(\Theta) = \frac{\text{adjacent}}{\text{hypotenuse}} side\]

\[\frac{b}{a} = Tan(\Theta) = \frac{\text{opposite}}{\text{adjacent}} side\]

We can also define their reciprocals as 

\[\frac{h}{b} = \frac{1}{sin(\Theta)} = cosec(\Theta) = \frac{\text{hypotenuse}}{\text{opposite}} side\]

\[\frac{h}{a} = \frac{1}{cos(\Theta)} = sec(\Theta) = \frac{\text{hypotenuse}}{\text{adjacent}} side\]

\[\frac{a}{b} = \frac{1}{tan(\Theta)} = cot(\Theta) = \frac{\text{adjacent}}{\text{opposite}} side\]

Out of these cosine(θ) ,sine (θ) and tan(θ) are more important trigonometric angles.

Here θ is usually expressed in radians. 

180° = 𝜋 radians

Two Important Right Triangles

Suppose we have an equilateral triangle of a side of length 1 unit. Drawing an angle bisector towards the base will equally divide the base into ½ unit lengths. We have now obtained two right angles with hypotenuse equal to 1 unit and the base equal to ½ units. We also know that the angle opposite to hypotenuse is 900 and the angle opposite to altitude is 600. Hence it is clear that the other angle is 300. Using the Pythagoras theorem we obtain the value of altitude as \[\frac{\sqrt{3}}{2}\] units.

If we try to define the above ratios for the two angles we obtain

  • Cos(60) = cos(𝜋/3) = \[\frac{\sqrt{3}}{2}\]

  • Sin(60) =  sin(𝜋/3) = \[\frac{1}{2}\]

  • Tan(60) = tan(𝜋/3) = \[\frac{1}{\sqrt{3}}\]

Similarly, we can write,

  • Cos(30) =cos(𝜋/6) = \[\frac{1}{2}\]

  • Sin(30) = sin(𝜋/6) = \[\frac{\sqrt{3}}{2}\]

  • Tan(30) = tan(𝜋/6) = \[\sqrt{3}\]

Let us do the same with an isosceles triangle with two angles equal to 450  two sides equal to 1. We obtain two right triangles with two angles equal to 450. Here we obtain

  • Cos(45) =cos(𝜋/4) = \[\frac{1}{\sqrt{2}}\]

  • Sin(45) = sin(𝜋/4) = \[\frac{1}{\sqrt{2}}\]

  • Tan(45) = tan(𝜋/4) = 1

These values are really helpful in various calculations. Similarly, there are few trigonometric angles which if known can predict the nature of solutions to various problems.

FAQ (Frequently Asked Questions)

1.What Are Allied Angles in Trigonometry? Write Some Properties of the Trigonometric Angles.

Observing the table, we can see that sin (30) is equal to cos (60) and vice versa. Looking more clearly we can understand that this kind of relationship exists if the sum or difference between two angles is 900. These types of angles are called allied angles.

Below are Some Properties of Trigonometric Angles.

  • Sin(θ) = - sin(θ)

  • Cos(-θ) = cos(θ)

  • Tan(θ) = - tan(θ)

  • Sin2(θ) + cos2(θ) = 1

  • 1+tan2(θ) = sec2(θ)

  • 1+cot2(θ) = cosec2(θ)