key term - Sin²θ + cos²θ = 1

Definition

The trigonometric identity $sin^2\theta + cos^2\theta = 1$ is a fundamental relationship between the sine and cosine functions. It states that the sum of the squares of the sine and cosine of any angle is always equal to 1. This identity is crucial for verifying trigonometric identities and simplifying trigonometric expressions.

5 Must Know Facts For Your Next Test

1. The identity $sin^2\theta + cos^2\theta = 1$ holds true for any angle $\theta$ measured in radians or degrees.
2. This identity is derived from the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
3. The identity $sin^2\theta + cos^2\theta = 1$ can be used to verify other trigonometric identities by manipulating the expressions and simplifying them.
4. Knowing this identity can help simplify trigonometric expressions by substituting the appropriate trigonometric functions.
5. The identity is often used in calculus and other advanced mathematics to derive and prove additional trigonometric relationships.

Review Questions

• Explain how the identity $sin^2\theta + cos^2\theta = 1$ is derived from the Pythagorean theorem.
  • The identity $sin^2\theta + cos^2\theta = 1$ is derived from the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In the unit circle, the sine and cosine values represent the $x$ and $y$ coordinates of a point on the circle, respectively. Therefore, the sum of the squares of these coordinates must be equal to the square of the radius of the circle, which is 1.
• Describe how the identity $sin^2\theta + cos^2\theta = 1$ can be used to verify other trigonometric identities.
  • The identity $sin^2\theta + cos^2\theta = 1$ can be used to verify other trigonometric identities by manipulating the expressions and simplifying them. For example, if you have a trigonometric identity that involves the sine and cosine functions, you can use this identity to simplify the expression and show that it is equal to 1. This can help confirm the validity of the original identity and demonstrate the relationships between the trigonometric functions.
• Explain how the identity $sin^2\theta + cos^2\theta = 1$ can be used to simplify trigonometric expressions.
  • The identity $sin^2\theta + cos^2\theta = 1$ can be used to simplify trigonometric expressions by substituting the appropriate trigonometric functions. For instance, if you have an expression that contains both $sin\theta$ and $cos\theta$, you can use this identity to replace one of the terms with the other, making the expression simpler and easier to evaluate. This can be particularly useful in calculus and other advanced mathematics when dealing with complex trigonometric functions and expressions.

Related terms

Trigonometric Functions:

The basic trigonometric functions are sine (sin), cosine (cos), and tangent (tan), which describe the ratios between the sides of a right triangle.

Unit Circle:

A circular representation of the trigonometric functions, where the coordinates of a point on the circle correspond to the sine and cosine values of the angle.

Pythagorean Theorem:

A fundamental theorem in geometry stating that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.