Unit Circle and Trigonometric Identities

The unit circle identities of sine, cosecant, and tangent can be further used to obtain the other trigonometric identities such as cotangent, secant, and cosecant. The unit circle identities such as cosecant, secant, cotangent are the respective reciprocal of the sine, cosine, tangent. Further, we can obtain the value of tanθ by dividing sinθ with cosθ, and we can obtain the value of cotθ by dividing cosθ with sinθ.

For a right triangle placed in a unit circle in the cartesian coordinate plane, with hypotenuse, base, and altitude measuring 1, x, y units respectively, the unit circle identities can be given as,

• sinθ = y/1
• cosθ = x/1
• tanθ = sinθ/cosθ = y/x
• sec(θ = 1/x
• csc(θ) = 1/y
• cot(θ) = cosθ/sinθ = x/y

Unit Circle Pythagorean Identities

The three important Pythagorean identities of trigonometric ratios can be easily understood and proved with the unit circle. The Pythagoras theorem states that in a right-angled triangle the square of the hypotenuse is equal to the sum of the square of the other two sides. The three Pythagorean identities in trigonometry are as follows.

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = cosec²θ

Here we shall try to prove the first identity with the help of the Pythagoras theorem. Let us take x and y as the legs of the right-angled triangle having a hypotenuse 1 unit. Applying Pythagoras theorem we have x² + y² = 1 which represents the equation of a unit circle. Also in a unit circle, we have, x = cosθ, and y = sinθ, and applying this in the above statement of the Pythagoras theorem, we have, cos²θ + sin²θ = 1. Thus we have successfully proved the first identity using the Pythagoras theorem. Further within the unit circle, we can also prove the other two Pythagorean identities.