bobsmith76 said:
Yes, draw a right triangle and label one of the angles x. Now label each side a, b and c. Ok so what is sin(x) in terms of a,b,c? So what is sin2(x)? Continue this for cos2(x) and you'll see the result holds.bobsmith76 said:
Basic trigonometry? After being taught about graphing trig functions I believe you're exposed to more trig identities.bobsmith76 said:
bobsmith76 said:
Yes, the identity 1 - sin²x = cos²x is true. This is known as the Pythagorean trigonometric identity, which relates the sine and cosine of the same angle.
This identity is derived from the Pythagorean theorem applied to the unit circle. In the unit circle, for any angle \( x \), the sine represents the y-coordinate, and the cosine represents the x-coordinate. According to the Pythagorean theorem, \( \sin^2x + \cos^2x = 1 \), and rearranging this gives \( 1 - \sin^2x = \cos^2x \).
This identity is useful in solving trigonometric equations, simplifying expressions, and proving other trigonometric identities. It's a fundamental identity in trigonometry.
Yes, the identity 1 - sin²x = cos²x holds true for all real values of \( x \). It is universally applicable in trigonometry, regardless of the angle's size or quadrant.
Yes, similar identities exist for other trigonometric functions. For example, the identities relating to the tangent and secant functions, such as \( 1 + \tan^2x = \sec^2x \), are based on similar principles.
The identity also illustrates the complementary relationship between sine and cosine: \( \sin x = \cos(90^\circ - x) \). This is because \( \sin^2x + \cos^2(90^\circ - x) = 1 \) simplifies to the same identity.
Yes, understanding this identity is important in calculus, especially when integrating or differentiating trigonometric functions. It aids in simplifying expressions before applying calculus techniques.