Trigonometric Identities Calculator

Calculate all six trigonometric function values from a single known value

Select a known trigonometric function, enter its value, and choose the quadrant to find the values of all other functions.

Examples

Click on any example to load it into the calculator.

Given sin(θ) in Q2

sin_q2

Find all trig values when sin(θ) is 0.5 in Quadrant II.

Function: SIN(θ)

Value: 0.5

Quadrant: 2

Given cos(θ) in Q4

cos_q4

Find all trig values when cos(θ) is 0.8 in Quadrant IV.

Function: COS(θ)

Value: 0.8

Quadrant: 4

Given tan(θ) in Q3

tan_q3

Find all trig values when tan(θ) is 1.2 in Quadrant III.

Function: TAN(θ)

Value: 1.2

Quadrant: 3

Given sec(θ) in Q1

sec_q1

Find all trig values when sec(θ) is 2 in Quadrant I.

Function: SEC(θ)

Value: 2

Quadrant: 1

Other Titles
Understanding Trigonometric Identities: A Comprehensive Guide
Explore the fundamental relationships between trigonometric functions and how to use them to solve mathematical problems.

What are Trigonometric Identities?

  • Fundamental equations relating trigonometric functions
  • True for all values of the involved variables
  • The building blocks of trigonometry and calculus
Trigonometric identities are equations involving trigonometric functions that hold true for all possible values of the input angle (θ). They are fundamental tools used to simplify complex trigonometric expressions, solve equations, and are essential in calculus for integrating non-standard functions.
Core Identity Categories
Reciprocal Identities: These define the relationships between pairs of functions. (e.g., csc(θ) = 1/sin(θ)).
Quotient Identities: These express one function in terms of two others. (e.g., tan(θ) = sin(θ)/cos(θ)).
Pythagorean Identities: Derived from the Pythagorean theorem on the unit circle, these are some of the most important identities (e.g., sin²(θ) + cos²(θ) = 1).

Fundamental Identity Examples

  • sin²(θ) + cos²(θ) = 1
  • 1 + tan²(θ) = sec²(θ)
  • tan(θ) = sin(θ) / cos(θ)

Step-by-Step Guide to Using the Trig Identities Calculator

  • Select your known function and its value
  • Specify the correct quadrant for the angle
  • Interpret the complete set of results
Our calculator simplifies the process of finding all trigonometric values from a single piece of information. Follow these steps for an accurate calculation.
Input Guidelines:
1. Select Known Function: Choose the function (sin, cos, tan, etc.) for which you have a value from the dropdown menu.
2. Enter Function Value: Input the numeric value corresponding to your chosen function. Ensure it's within the valid range (e.g., -1 to 1 for sin and cos).
3. Choose the Quadrant: Select the quadrant (I, II, III, or IV) in which the angle θ lies. This is crucial as it determines the positive or negative sign of the other functions. The 'ASTC' rule (All, Sine, Tangent, Cosine) can help remember which functions are positive in each quadrant.
Interpreting Results:
The calculator will output the values for all six trigonometric functions based on your inputs, as well as the principal angle in both degrees and radians.

Practical Usage Examples

  • Input: sin(θ) = 0.5, Quadrant II -> Output: cos(θ) = -0.866, tan(θ) = -0.577, etc.
  • Input: tan(θ) = -1, Quadrant IV -> Output: sin(θ) = -0.707, cos(θ) = 0.707, etc.

Real-World Applications of Trigonometric Identities

  • Physics and Engineering: Analyzing waves, oscillations, and AC circuits
  • Computer Graphics: 3D modeling, rotations, and transformations
  • Navigation and Astronomy: Calculating positions and distances
Trigonometric identities are not just abstract mathematical concepts; they are vital in numerous scientific and technological fields.
Engineering and Physics:
In electrical engineering, trigonometric identities are used to analyze alternating current (AC) circuits. In mechanical engineering and physics, they describe harmonic motion and wave phenomena, from swinging pendulums to light and sound waves.
Digital and Creative Fields:
In computer graphics, identities are used to calculate rotations and transformations of objects in 3D space. Game developers and animators rely on them heavily. In music synthesis, trigonometric functions model sound waves, and identities help in manipulating them to create different sounds.
Navigation and Surveying:
GPS systems, astronomy, and land surveying use triangulation, a method that relies heavily on trigonometric relationships to determine distances and positions.

Industry Applications

  • Modeling the oscillation of a spring using sine and cosine.
  • Calculating the angle of a light ray refracting through a prism.
  • Rotating a character in a video game.

Common Misconceptions and Correct Methods

  • Understanding the importance of the quadrant
  • Correctly applying the Pythagorean identities
  • Avoiding common algebraic errors
Solving trigonometric problems requires careful attention to detail. Here are some common pitfalls and how to avoid them.
The Quadrant is Not Optional
A common mistake is ignoring the quadrant. For example, if cos²(θ) = 0.25, then cos(θ) could be 0.5 or -0.5. The quadrant determines the correct sign. Cosine is positive in Quadrants I and IV, and negative in II and III. Forgetting this leads to incorrect results for all other functions.
Square Root Ambiguity
When using a Pythagorean identity like sin²(θ) = 1 - cos²(θ), you must take a square root. Remember that √(x²) = |x|, not just x. The sign is then determined by the quadrant, as mentioned above.
Range Errors
A value of sin(θ) or cos(θ) must be between -1 and 1, inclusive. Likewise, a value for csc(θ) or sec(θ) must be >= 1 or <= -1. Entering a value outside this range is a common error.

Error-Checking Examples

  • Incorrect: sin(θ) = 0.8 in Q4 (Sine is negative in Q4).
  • Correct: sin(θ) = -0.8 in Q4.
  • Incorrect: cos(θ) = 1.5 (Value out of range).
  • Correct: cos(θ) = 0.5.

Mathematical Derivation and Formulas

  • Deriving identities from the unit circle
  • Step-by-step calculation from a known value
  • Formulas for each function derivation
The core of these calculations lies in the Pythagorean Identities, derived from a right-angled triangle in the unit circle with hypotenuse 1.
Example Derivation: Given sin(θ)
1. Find cos(θ): Use cos²(θ) + sin²(θ) = 1. So, cos(θ) = ±√(1 - sin²(θ)). The sign is chosen based on the quadrant.
2. Find tan(θ): Use tan(θ) = sin(θ) / cos(θ).
3. Find Reciprocals: csc(θ) = 1/sin(θ), sec(θ) = 1/cos(θ), cot(θ) = 1/tan(θ).
Example Derivation: Given tan(θ)
1. Find sec(θ): Use sec²(θ) = 1 + tan²(θ). So, sec(θ) = ±√(1 + tan²(θ)). The sign is chosen based on the quadrant.
2. Find cos(θ): Use cos(θ) = 1 / sec(θ).
3. Find sin(θ): Use sin(θ) = tan(θ) * cos(θ).
4. Find Reciprocals: csc(θ) = 1/sin(θ), cot(θ) = 1/tan(θ).

Calculation Formulas

  • If sin(θ) = 3/5 in Q1, cos(θ) = √(1 - (3/5)²) = 4/5.
  • If tan(θ) = -5/12 in Q2, sec(θ) = -√(1 + (-5/12)²) = -13/12.