Trigonometric Functions and Pythagorean Identities

Trigonometric Functions and Identities

Relationships Between Trigonometric Functions

  • Six trigonometric functions are interconnected.
  • Reciprocal relationships:
    • csc⁡(θ)=1sin⁡(θ)\csc(\theta) = \frac{1}{\sin(\theta)}
    • sec⁡(θ)=1cos⁡(θ)\sec(\theta) = \frac{1}{\cos(\theta)}
    • cot⁡(θ)=1tan⁡(θ)\cot(\theta) = \frac{1}{\tan(\theta)}
  • Tangent in terms of sine and cosine: tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} (y-coordinate divided by x-coordinate on the unit circle).

The Unit Circle and Pythagorean Identity

  • Equation of the unit circle: x2+y2=1x^2 + y^2 = 1
  • For a point on the unit circle corresponding to angle θ\theta:
    • x-coordinate: cos⁡(θ)\cos(\theta)
    • y-coordinate: sin⁡(θ)\sin(\theta)
  • Therefore, (cos⁡(θ))2+(sin⁡(θ))2=1(\cos(\theta))^2 + (\sin(\theta))^2 = 1

Notation

  • Shorthand notation:
    • cos⁡2(θ)\cos^2(\theta) means (cos⁡(θ))2(\cos(\theta))^2 (find cosine of theta, then square).
    • sin⁡2(θ)\sin^2(\theta) means (sin⁡(θ))2(\sin(\theta))^2 (find sine of theta, then square).
  • Important distinction:
    • sin⁡2(θ)\sin^2(\theta) is not the same as sin⁡(θ2)\sin(\theta^2)

Pythagorean Identity

  • The identity: cos⁡2(θ)+sin⁡2(θ)=1\cos^2(\theta) + \sin^2(\theta) = 1
  • Based on the Pythagorean theorem applied to the unit circle: (adjacent)2^2 + (opposite)2^2 = (hypotenuse)2^2, with hypotenuse = 1.

Applications of the Pythagorean Identity

  • If the cosine of an angle is known, the sine can be found (within a ±\pm sign), and vice versa.

Alternative Forms of the Pythagorean Identity

  • Dividing the original identity by sin⁡2(θ)\sin^2(\theta):
    • cos⁡2(θ)sin⁡2(θ)+sin⁡2(θ)sin⁡2(θ)=1sin⁡2(θ)\frac{\cos^2(\theta)}{\sin^2(\theta)} + \frac{\sin^2(\theta)}{\sin^2(\theta)} = \frac{1}{\sin^2(\theta)}
    • cot⁡2(θ)+1=csc⁡2(θ)\cot^2(\theta) + 1 = \csc^2(\theta)
  • Dividing the original identity by cos⁡2(θ)\cos^2(\theta):
    • cos⁡2(θ)cos⁡2(θ)+sin⁡2(θ)cos⁡2(θ)=1cos⁡2(θ)\frac{\cos^2(\theta)}{\cos^2(\theta)} + \frac{\sin^2(\theta)}{\cos^2(\theta)} = \frac{1}{\cos^2(\theta)}
    • 1+tan⁡2(θ)=sec⁡2(θ)1 + \tan^2(\theta) = \sec^2(\theta)

Naming Conventions

  • The "co" in trigonometric function names (cosine, cotangent, cosecant) follows a pattern in the identities: "co" goes with "co", and "not co" goes with "not co".
    • 1+tan⁡2(θ)=sec⁡2(θ)1 + \tan^2(\theta) = \sec^2(\theta)
    • 1+cot⁡2(θ)=csc⁡2(θ)1 + \cot^2(\theta) = \csc^2(\theta)

Using Trigonometric Identities to Solve Problems

  • If some information about an angle is known, trigonometric identities can be used to determine other trigonometric function values.

Example

  • Given: Angle θ\theta in the second quadrant, and cos⁡(θ)=−0.3\cos(\theta) = -0.3
  • Goal: Find sin⁡(θ)\sin(\theta), tan⁡(θ)\tan(\theta), cot⁡(θ)\cot(\theta), sec⁡(θ)\sec(\theta), and csc⁡(θ)\csc(\theta).
  • Using the Pythagorean identity: cos⁡2(θ)+sin⁡2(θ)=1\cos^2(\theta) + \sin^2(\theta) = 1
    • (−0.3)2+sin⁡2(θ)=1(-0.3)^2 + \sin^2(\theta) = 1
    • 0.09+sin⁡2(θ)=10.09 + \sin^2(\theta) = 1
    • sin⁡2(θ)=1−0.09=0.91\sin^2(\theta) = 1 - 0.09 = 0.91
    • sin⁡(θ)=±0.91\sin(\theta) = \pm\sqrt{0.91}
  • Since θ\theta is in the second quadrant, sin⁡(θ)\sin(\theta) is positive. Therefore, sin⁡(θ)=0.91\sin(\theta) = \sqrt{0.91}.
  • Now that both sin⁡(θ)\sin(\theta) and cos⁡(θ)\cos(\theta) are known, the other trigonometric functions can be found using their definitions.
    • tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
    • cot⁡(θ)=1tan⁡(θ)\cot(\theta) = \frac{1}{\tan(\theta)}
    • sec⁡(θ)=1cos⁡(θ)\sec(\theta) = \frac{1}{\cos(\theta)}
    • csc⁡(θ)=1sin⁡(θ)\csc(\theta) = \frac{1}{\sin(\theta)}

Key Takeaways

  • The Pythagorean identity (and its variations) is crucial for relating trigonometric functions.
  • Knowing one trigonometric function value often allows you to find all the others.