Trigonometric Identities Calculator
Calculate unknown trigonometric function values using fundamental identities (Pythagorean, Sum/Difference, Double/Half Angle) when one value is known. Perfect for learning and verifying trigonometric relationships!
About Trigonometric Identities Calculator
Welcome to our Trigonometric Identities Calculator, a powerful tool for calculating unknown trigonometric function values using fundamental identities. When you know one trigonometric value (like $\sin(x)$ or $\cos(x)$) and the quadrant, this calculator finds all other trigonometric values and applies various identities.
What This Calculator Does
Given a known trigonometric function value and the quadrant where the angle lies, this calculator:
- Calculates all six basic trig functions: $\sin(x)$, $\cos(x)$, $\tan(x)$, $\csc(x)$, $\sec(x)$, $\cot(x)$
- Applies Double Angle Formulas: Find $\sin(2x)$, $\cos(2x)$, and $\tan(2x)$
- Applies Half Angle Formulas: Find $\sin(x/2)$, $\cos(x/2)$, and $\tan(x/2)$
- Determines the exact angle: Both in degrees and radians
- Verifies identities: Confirms the Pythagorean identity $\sin^2(x) + \cos^2(x) = 1$
Key Trigonometric Identities
- $$\sin^2(x) + \cos^2(x) = 1$$
- $$1 + \tan^2(x) = \sec^2(x)$$
- $$1 + \cot^2(x) = \csc^2(x)$$
- $$\sin(2x) = 2\sin(x)\cos(x)$$
- $$\cos(2x) = \cos^2(x) - \sin^2(x) = 2\cos^2(x) - 1 = 1 - 2\sin^2(x)$$
- $$\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}$$
- $$\sin\left(\frac{x}{2}\right) = \pm\sqrt{\frac{1 - \cos(x)}{2}}$$
- $$\cos\left(\frac{x}{2}\right) = \pm\sqrt{\frac{1 + \cos(x)}{2}}$$
- $$\tan\left(\frac{x}{2}\right) = \frac{\sin(x)}{1 + \cos(x)} = \frac{1 - \cos(x)}{\sin(x)}$$
Understanding Quadrants
The sign of each trigonometric function depends on which quadrant the angle is in:
- Quadrant I (0° - 90°): All functions are positive (A)
- Quadrant II (90° - 180°): Only sin and csc are positive (S)
- Quadrant III (180° - 270°): Only tan and cot are positive (T)
- Quadrant IV (270° - 360°): Only cos and sec are positive (C)
Remember: "All Students Take Calculus" or "ASTC"
Reciprocal Identities
- $\csc(x) = \frac{1}{\sin(x)}$
- $\sec(x) = \frac{1}{\cos(x)}$
- $\cot(x) = \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)}$
How to Use This Calculator
- Select the known function: Choose which trigonometric function value you know (sin, cos, tan, csc, sec, or cot).
- Enter the known value: Input the numerical value of the function.
- Select the quadrant: Choose which quadrant the angle x lies in (this determines the signs of other functions).
- Choose identity type: Select which identities to apply (Pythagorean, Double Angle, Half Angle, or All).
- Set precision: Choose how many decimal places you want (1-100).
- Click Calculate: View the complete results with step-by-step solutions.
Applications
- Education: Learn and verify trigonometric identities
- Engineering: Signal processing and wave analysis
- Physics: Harmonic motion and oscillation problems
- Navigation: Calculating angles and distances
- Computer Graphics: Rotation and transformation matrices
Additional Resources
Reference this content, page, or tool as:
"Trigonometric Identities Calculator" at https://MiniWebtool.com/trigonometric-identities-calculator/ from MiniWebtool, https://MiniWebtool.com/
by miniwebtool team. Updated: Nov 24, 2025
You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.
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