6.12 Integrating Using Trigonometric Identities

• What are trigonometric identities used for in integration?

  Simplifying trigonometric integrals
• The double-angle identity for $\sin(2x)$ is $2\sin(x)\cos(x)$, which is useful for simplifying integrals with $\sin(2x)$ or dealing with $\sin(x)\cos(x)$ terms.double-angle
• The Pythagorean identity is $\sin^{2}(x) +$$\cos^{2}(x) =$$1$.
• The half-angle identity for $\sin^{2}(x)$ is $\frac{1 - \cos(2x)}{2}$, which is useful for reducing powers of $\sin(x)$ in integrals.half-angle
• Using trigonometric identities can simplify complex integrals into forms easily solved using basic integration techniques from Unit 6.4.
• What is the result of integrating $\sin(2x)$ using its double-angle identity?


  $- \frac{\cos(2x)}{2} +$$C$
• The Pythagorean identity can be used to convert $\cos^{2}(x)$ into 1 - \sin^{2}(x)</latex>, which reduces the complexity of the integral.Pythagorean
• The integral of $\sin^{2}(x)$ using its half-angle identity is $\frac{x}{2} - \frac{\sin(2x)}{4} +$$C$.
• What is the first step in integrating $\sin(2x)$ using its double-angle identity?


  Rewrite as $2\sin(x)\cos(x)$
• The integral of $\cos^{2}(x)$ is $x - \frac{1}{3}\sin^{3}(x) +$$C$ after converting it using the Pythagorean identity.
• What identity is used to reduce the power of $\cos(x)$ in integrals?


  Half-angle identity
• What is the double-angle formula for $\sin(2x)$?


  $\sin(2x) =$$2\sin(x)\cos(x)$
• The Pythagorean identity states that $\sin^{2}(x) +$$\cos^{2}(x) =$$1$.
• What is the half-angle formula for $\sin^{2}(x)$?


  \sin^{2}(x) = \frac{1 - \cos(2x)}{2}</latex>
• Trigonometric identities are used to simplify integrals involving trigonometric functions
• What is the double-angle formula for $\sin(2x)$?


  \sin(2x) = 2\sin(x)\cos(x)</latex>
• The Pythagorean identity states that $\sin^{2}(x) +$$\cos^{2}(x) =$$1$.
• What is the half-angle formula for $\cos^{2}(x)$?


  $\cos^{2}(x) =$$\frac{1 + \cos(2x)}{2}$
• Match the trigonometric identity with its formula:

  Double-Angle ↔️ \sin(2x) = 2\sin(x)\cos(x)</latex>

  Pythagorean ↔️ $\sin^{2}(x) +$$\cos^{2}(x) =$$1$

  Half-Angle ↔️ $\sin^{2}(x) =$$\frac{1 - \cos(2x)}{2}$
• The double-angle formula for $\sin(2x)$ is used to simplify $\sin(2x)$ or $\sin(x)\cos(x)$.
• What is the Pythagorean identity?

  \sin^{2}(x) + \cos^{2}(x) = 1</latex>
• The Pythagorean identity is used to convert between sine and cosine
• The Pythagorean identity can be written as $\cos^{2}(x) =$$1 - \sin^{2}(x)$.
• What is the purpose of the half-angle formula for $\sin^{2}(x)$?


  Reduce powers of $\sin(x)$
• The half-angle formula for $\sin^{2}(x)$ is $\sin^{2}(x) =$$\frac{1 - \cos(2x)}{2}$.
• What is the result of $\int \sin^{2}(x) dx$ using the half-angle formula?


  $\frac{x}{2} - \frac{\sin(2x)}{4} +$$C$
• The half-angle formula can be used to simplify the integral $\int \sin^{2}(x) dx$.
• How does the double-angle formula $\sin(2x) =$$2\sin(x)\cos(x)$ simplify integrals?


  Simplifies $\sin(2x)$ or $\sin(x)\cos(x)$
• The Pythagorean identity is \sin^{2}(x) + \cos^{2}(x) = 1
• The half-angle identity for $\sin^{2}(x)$ is \frac{1 - \cos(2x)}{2}</latex>
• What is the double-angle formula for $\sin(2x)$?


  2\sin(x)\cos(x)
• The Pythagorean identity is used to convert between sine and cosine.
• The half-angle identity $\sin^{2}(x) =$$\frac{1 - \cos(2x)}{2}$ reduces the power of $\sin(x)$
• Using the half-angle identity, \int \sin^{2}(x) dx = \frac{x}{2} - \frac{\sin(2x)}{4} + C
• What is the integral of \sin(2x)</latex> using the double-angle identity?

  \frac{\cos(2x)}{2} + C
• Converting $\cos^{2}(x)$ to $(1 - \sin^{2}(x))$ simplifies the integral of $\cos^{2}(x)$
• The integral of $\sin^{2}(x)$ using the half-angle identity is \frac{x}{2} - \frac{\sin(2x)}{4} + C
• Match the identity with its use:

  Double-Angle ↔️ Simplify $\sin(2x)$

  Pythagorean ↔️ Convert between sine and cosine

  Half-Angle ↔️ Reduce powers of $\sin(x)$
• What is the double-angle identity for $\sin(2x)$?


  $\sin(2x) =$$2\sin(x)\cos(x)$
• The Pythagorean identity is $\sin^{2}(x) +$$\cos^{2}(x) =$$1$