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Calculus 1Calculus 126 views·Updated Jul 20, 2026·2 pages

Trigonometric Identities for Calculus and Pre-Calculus

user profile picture
Mallory Joyce@alloryoyce_lxwolbffx

Trigonometric identities are essential formulas that show relationships between trigonometric...

1
of 2
Trig identities: calculus and pre-calculus – page 1

Angle Identities

The angle sum identities let you break down complex angles into simpler parts. When working with sin(A+B)sin(A+B), you can rewrite it as sin(A)cos(B)+cos(A)sin(B)sin(A)cos(B) + cos(A)sin(B). Similarly, cos(A+B)cos(A+B) equals cos(A)cos(B)−sin(A)sin(B)cos(A)cos(B) - sin(A)sin(B).

For angle differences, the formulas are slightly modified. The sin(A−B)sin(A-B) formula becomes sin(A)cos(B)−cos(A)sin(B)sin(A)cos(B) - cos(A)sin(B), while cos(A−B)cos(A-B) equals cos(A)cos(B)+sin(A)sin(B)cos(A)cos(B) + sin(A)sin(B).

Double angle identities are particularly useful shortcuts. You can express sin(2A)sin(2A) as 2sinAcosA2sinAcosA. For cosine, you have three equivalent options: cos(2A)=cos2A−sin2Acos(2A) = cos^{2}A - sin^{2}A, or cos(2A)=1−2sin2Acos(2A) = 1 - 2sin^{2}A.

Pro Tip: Memorize these identities by understanding patterns rather than rote memorization. Notice how the angle sum and difference formulas only change signs!

2
of 2
Trig identities: calculus and pre-calculus – page 2

More Double and Half Angle Formulas

Another useful way to write the double angle formula is cos(2A)=2cos2A−1cos(2A) = 2cos^2A - 1. This version is particularly helpful when you're already working with cosine values and want to avoid converting to sine.

Half angle formulas let you find trig values of angles like 15° or 22.5°. For cosine, use cos(A2)=±(1+cosA2)cos(\frac{A}{2}) = \pm \sqrt{(\frac{1 + cosA}{2})}. Choose the positive sign when A2\frac{A}{2} is in the first or fourth quadrant, and negative otherwise.

For sine half angles, the formula is sin(A2)=±(1−cosA2)sin(\frac{A}{2}) = \pm \sqrt{(\frac{1 - cosA}{2})}. The sign is positive when A2\frac{A}{2} is in the first or second quadrant, and negative otherwise.

Remember: Half angle formulas always include a ± sign. The correct choice depends on which quadrant your angle falls in!

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Calculus 1Calculus 126 views·Updated Jul 20, 2026·2 pages

Trigonometric Identities for Calculus and Pre-Calculus

user profile picture
Mallory Joyce@alloryoyce_lxwolbffx

Trigonometric identities are essential formulas that show relationships between trigonometric functions. These powerful shortcuts help solve complex problems and are crucial for advanced math courses like calculus and physics.

1
of 2
Trig identities: calculus and pre-calculus – page 1

Sign up to see the content. It's free!

  • Access to all documents
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Angle Identities

The angle sum identities let you break down complex angles into simpler parts. When working with sin(A+B)sin(A+B), you can rewrite it as sin(A)cos(B)+cos(A)sin(B)sin(A)cos(B) + cos(A)sin(B). Similarly, cos(A+B)cos(A+B) equals cos(A)cos(B)−sin(A)sin(B)cos(A)cos(B) - sin(A)sin(B).

For angle differences, the formulas are slightly modified. The sin(A−B)sin(A-B) formula becomes sin(A)cos(B)−cos(A)sin(B)sin(A)cos(B) - cos(A)sin(B), while cos(A−B)cos(A-B) equals cos(A)cos(B)+sin(A)sin(B)cos(A)cos(B) + sin(A)sin(B).

Double angle identities are particularly useful shortcuts. You can express sin(2A)sin(2A) as 2sinAcosA2sinAcosA. For cosine, you have three equivalent options: cos(2A)=cos2A−sin2Acos(2A) = cos^{2}A - sin^{2}A, or cos(2A)=1−2sin2Acos(2A) = 1 - 2sin^{2}A.

Pro Tip: Memorize these identities by understanding patterns rather than rote memorization. Notice how the angle sum and difference formulas only change signs!

2
of 2
Trig identities: calculus and pre-calculus – page 2

Sign up to see the content. It's free!

  • Access to all documents
  • Improve your grades
  • Join milions of students

More Double and Half Angle Formulas

Another useful way to write the double angle formula is cos(2A)=2cos2A−1cos(2A) = 2cos^2A - 1. This version is particularly helpful when you're already working with cosine values and want to avoid converting to sine.

Half angle formulas let you find trig values of angles like 15° or 22.5°. For cosine, use cos(A2)=±(1+cosA2)cos(\frac{A}{2}) = \pm \sqrt{(\frac{1 + cosA}{2})}. Choose the positive sign when A2\frac{A}{2} is in the first or fourth quadrant, and negative otherwise.

For sine half angles, the formula is sin(A2)=±(1−cosA2)sin(\frac{A}{2}) = \pm \sqrt{(\frac{1 - cosA}{2})}. The sign is positive when A2\frac{A}{2} is in the first or second quadrant, and negative otherwise.

Remember: Half angle formulas always include a ± sign. The correct choice depends on which quadrant your angle falls in!

We thought you’d never ask...

Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.

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