Learning Outcomes
- Apply the Pythagorean identities
- Apply the product-to-sum formulas
- Use substitution to evaluate indefinite integrals containing trigonometric functions
In the Trigonometric Integrals section, we will learn how to evaluate integrals that contain trigonometric functions raised to powers. Here we will review trigonometric identifies and how to use substitution to evaluate trigonometric integrals.
Apply the Pythagorean Identities
Pythagorean identities are equations involving trigonometric functions based on the properties of a right triangle.
| Pythagorean Identities | ||
|---|---|---|
| [latex]{\sin }^{2}\theta +{\cos }^{2}\theta =1[/latex] | [latex]1+{\tan }^{2}\theta ={\sec }^{2}\theta[/latex] | [latex]1+{\cot }^{2}\theta ={\csc }^{2}\theta[/latex] |
- [latex]{\cos }^{2}\theta =1-{\sin }^{2}\theta [/latex]
- [latex]{\sin }^{2}\theta =1 - {\cos }^{2}\theta[/latex]
- [latex]{\tan }^{2}\theta = {\sec }^{2}\theta - 1[/latex]
- [latex]{\sec }^{2}\theta = 1+{\tan }^{2}\theta[/latex] (same as the original identity but sides of the equation are swapped)
Example: Using Pythagorean Identities to Rewrite Expressions
Simplify [latex]\frac{{\sec }^{2}\theta -1}{{\sec }^{2}\theta }[/latex].
Example: Using Pythagorean Identities to Rewrite Expressions
Rewrite the expression [latex]{\sin}\theta {\cos^4}\theta[/latex] using the identity [latex]{\cos }^{2}\theta =1 - {\sin }^{2}\theta[/latex].
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Rewrite the expression [latex]{\tan^4}\theta[/latex] using the identity [latex]{\tan }^{2}\theta = {\sec }^{2}\theta - 1[/latex].
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Rewrite the expression [latex]{\sin^2}\theta {\cos}\theta[/latex] using the identity [latex]{\sin }^{2}\theta =1 - {\cos }^{2}\theta[/latex].
Apply the Product-to-Sum Formulas
Sometimes, we may need to express the product of cosine and sine as a sum. We can use the product-to-sum formulas, which express products of trigonometric functions as sums. Note the following product-to-sum formulas.
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[latex]\cos \alpha \cos \beta =\frac{1}{2}\left[\cos \left(\alpha -\beta \right)+\cos \left(\alpha +\beta \right)\right][/latex]
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[latex]\sin \alpha \cos \beta =\frac{1}{2}\left[\sin \left(\alpha +\beta \right)+\sin \left(\alpha -\beta \right)\right][/latex]
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[latex]\sin \alpha \sin \beta =\frac{1}{2}\left[\cos \left(\alpha -\beta \right)-\cos \left(\alpha +\beta \right)\right][/latex]
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[latex]\cos \alpha \sin \beta =\frac{1}{2}\left[\sin \left(\alpha +\beta \right)-\sin \left(\alpha -\beta \right)\right][/latex]
Example: Expanding Using a Product-To-Sum Formula
Write the following product of cosines as a sum: [latex]2\cos \left(\frac{7x}{2}\right)\cos \left(\frac{3x}{2}\right)[/latex].
Example: Expanding Using a Product-To-Sum Formula
Express the following product as a sum containing only sine or cosine and no products: [latex]\sin \left(4\theta \right)\cos \left(2\theta \right)[/latex].
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Use the product-to-sum formula to write the product as a sum or difference: [latex]\cos \left(2\theta \right)\cos \left(4\theta \right)[/latex].
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Use Substitution to Evaluate Indefinite Integrals Containing Trigonometric Functions
We can generalize substitution using the following steps:
- Look carefully at the integrand and select an expression [latex]g(x)[/latex] within the integrand to set equal to [latex]u[/latex]. Let’s select [latex]g(x).[/latex] such that [latex]{g}^{\prime }(x)[/latex] is also part of the integrand.
- Substitute [latex]u=g(x)[/latex] and [latex]du={g}^{\prime }(x)dx.[/latex] into the integral.
- We should now be able to evaluate the integral with respect to [latex]u[/latex]. If the integral can’t be evaluated we need to go back and select a different expression to use as [latex]u[/latex].
- Evaluate the integral in terms of [latex]u[/latex].
- Write the result in terms of [latex]x[/latex] and the expression [latex]g(x).[/latex]
Example: Applying Substitution to Integrals with Trigonometric Functions
Use substitution to evaluate the integral [latex]\displaystyle\int \frac{ \sin t}{{ \cos }^{3}t}dt.[/latex]
Example: Applying Substitution to Integrals with Trigonometric Functions
Use substitution to evaluate the integral [latex]\displaystyle\int {\tan}(t) {\sec^2}(t) dt.[/latex]
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Use substitution to evaluate the integral [latex]\displaystyle\int \frac{ \cos t}{{ \sin }^{2}t}dt.[/latex]
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Use substitution to evaluate the indefinite integral [latex]\displaystyle\int { \cos }^{3}(t) \sin (t)dt.[/latex]
