Calculus 1

Dec 5, 2025

2 pages

Trigonometric Identities for Calculus and Pre-Calculus

Mallory Joyce @alloryoyce_lxwolbffx

Trigonometric identities are essential formulas that show relationships between trigonometric functions. These powerful shortcuts help solve complex problems... Show more

# Identities

- Angle Sum Identities:
$sin(A+B) = sin(A)cos(B) + cos(A)sin(B)$
$cos(A+B) = cos(A)cos(B) - sin(A)sin(B)$

- Angle Difference

Angle Identities

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

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

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

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

More Double and Half Angle Formulas

Another useful way to write the double angle formula is $cos(2A) = 2cos^2A - 1$ . 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 $cos(\frac{A}{2}) = \pm \sqrt{(\frac{1 + cosA}{2})}$ . Choose the positive sign when $\frac{A}{2}$ is in the first or fourth quadrant, and negative otherwise.

For sine half angles, the formula is $sin(\frac{A}{2}) = \pm \sqrt{(\frac{1 - cosA}{2})}$ . The sign is positive when $\frac{A}{2}$ 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 1

•

Dec 5, 2025

•

2 pages

Trigonometric Identities for Calculus and Pre-Calculus

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.

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Angle Identities

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

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

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

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More Double and Half Angle Formulas

For sine half angles, the formula is $sin(\frac{A}{2}) = \pm \sqrt{(\frac{1 - cosA}{2})}$ . The sign is positive when $\frac{A}{2}$ 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!