Equations-Angles Of Trigonometry

The word trigonometry is derived from the Greek words trigonon, which means “triangle”, and metron, which means “to measure”. Therefore, it deals with the measurement of the angles with the help of the sides of the triangle. Trigonometry consists of six specific functions: the ratios of the different sides of the triangles. These functions are:

• Sine(Sin),

• Cosine(Cos),

• Tangent(Tan),

• Cosecant(Cosec),

• Secant(Sec), and

• Cotangent(Cot)

Right-angled triangle

Trigonometric functions are the ratio of different sides of the triangle, but mostly it is only applicable to a right-angled triangle.

A right-angled triangle is a triangle that consists of an angle whose measure is 90°. The longest side of the right-angled triangle is known as the triangle’s hypotenuse.

Length of the Hypotenuse

The hypotenuse length can be found using the other two sides of the triangle by applying the Pythagorean theorem. According to the theorem, the sum of the square of the two sides of a right-angled triangle is equal to the square of the hypotenuse of the triangle.

Hypotenuse2=Base2+Perpendicular2

Ratios of the different sides of a triangle

When we calculate the value of the trigonometric functions, the side opposite to the angle is the perpendicular for that angle, while the side adjacent to the angle other than the hypotenuse is the base for that angle.

Sine(Sin)

The sine function of trigonometry is the ratio of the perpendicular height of the triangle to the hypotenuse of the triangle for that angle.

Sin(θ)=Perpendicular / Base

Cosine(Cos)

The cosine function is the ratio of the base of the triangle and the hypotenuse of the triangle for that angle

Cosθ=Base / Hypotenuse

Tangent(Tan)

The tangent is the ratio of the perpendicular height of the triangle to the base of the triangle. It is also equal to the ratio Sine and the cosine function.

Tan(θ)=Sine(θ) / Cosine(θ)

Tan()=Perpendicular/Hypotenuse/Base/Hypotenuse

Tan()=Perpendicular×Hypotenuse / Hypotenuse×Base

Tan(θ)=Perpendicular / Base

Cosecant(Cosec)

Cosecant is the ratio of the Hypotenuse of the triangle to the perpendicular of the triangle for that angle. It is also equal to the inverse of the sine function of the trigonometric function.

Cosec(θ)=1 / Sin(θ)

Cosec(θ)=1/Perpendicular/Hypotenuse

Cosec(θ)= Hypotenuse/ Perpendicular

Secant(Sec)

Secant is the ratio of the Hypotenuse of the triangle to the base of the triangle for that angle. It is also equal to the inverse of the Cosine function of the trigonometric function.

Sec(θ)=1 / Cos(θ)

Sec(θ)=1 / Base / Hypotenuse

Sec(θ)=Hypotenuse / Base

Cotangent(Cot)

Cotangent is the ratio of the base of the triangle to the perpendicular height of the triangle for that angle. It is also the inverse of the tangent function of the trigonometric function.

Cot(θ)=1 / Tan(θ)

Cot(θ)= 1 / Perpendicular / Base

Cot(θ)=Base / Perpendicular

Example

What is the length of the perpendicular side of the triangle if the length of the base of the triangle is 32 cm, and the angle of the triangle is 45°?

The length of the perpendicular side of the triangle is 32 cm.

Given to us

The length of the base of the triangle = 32 cm

The angle of the triangle= 45°

As discussed above, the Tangent is the ratio of the length of the perpendicular side of the triangle to the base of the triangle. Therefore, substitute the values in the formula to the tangent.

Tan(θ)=Perpendicular / Base

Tan(45o)=Perpendicular32

Tan(45o) ×32 =Perpendicular

1 ×32 = Perpendicular

Perpendicular = 32 cm.

Hence, the length of the perpendicular side of the triangle is 32 cm.

Values of the trigonometric function

The value of all the six different trigonometric functions is given in the table below for the measure of the angle 0°, 30°, 45°, 60°, 90°, 180°, 270°, and 360°.

Trigonometric Identities

Supplementary angles (sum =π)

• Sin (π–α) = sin α

• Cos (π–α) = – cos α

• Tan (π–α) = – tan α

• Cot (π–α) = – cot α

Anti – supplementary Angles (difference = π)

• Sin (π + α) = – sin α

• Cos (π + α) = – cos α

• Tan (π + α) = tan α

• Cot (π + α) = cot α

Opposite Angles (sum = 2π)

• Sin (2π–α) = – sin α

• Cos (2π–α) = cos α

• Tan (2π–α) = -tan α

• Cot (2π–α) = – cot α

Complementary Angles (sum = π/2)

• Sin (π/2 –α) = cos α

• Cos (π/2 –α) = sin α

• Tan (π/2 –α) = cot

• Cot (π/2 –α) = tan α

Conclusion

Trigonometric functions are ratios of the different sides of the triangle. It gives us a relationship between the different sides of the triangle and the angle of the triangle. 

There are six trigonometric ratios present which is interchangeable for different angles.