Find The Other Five Trigonometric Functions Of θ Calculator

Enter one trigonometric function value and the quadrant of θ to find the other five trigonometric functions.

Summary of Trigonometric Functions

Function	Value
sin(θ)	–
cos(θ)	–
tan(θ)	–
csc(θ)	–
sec(θ)	–
cot(θ)	–

Table: Values of the six trigonometric functions of θ.

What is a Find the Other Five Trigonometric Functions of θ Calculator?

A "Find the Other Five Trigonometric Functions of θ Calculator" is a tool used to determine the values of the five remaining trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) of an angle θ when the value of one of these functions and the quadrant in which θ lies are known. Trigonometric functions relate the angles of a triangle to the lengths of its sides, and they are fundamental in various fields like physics, engineering, navigation, and mathematics. This calculator is particularly useful for students learning trigonometry, engineers, and scientists who need to find all function values based on limited information.

Anyone studying or working with angles and their trigonometric ratios can use this calculator. Common users include high school and college students, math teachers, engineers, physicists, and architects. A common misconception is that knowing one function's value is enough; however, the quadrant is crucial because it determines the signs (positive or negative) of the other functions, as two different angles (in different quadrants) can have the same value for one trigonometric function (e.g., sin(30°) = 0.5 and sin(150°) = 0.5).

Find the Other Five Trigonometric Functions of θ Formula and Mathematical Explanation

The core idea is to use the given trigonometric function value and the quadrant to find the values of x, y, and r corresponding to a point (x, y) on the terminal side of the angle θ in standard position, where r is the distance from the origin to (x, y) (r = √(x² + y²), r > 0).

1. Identify Knowns: We know one function value (e.g., sin(θ) = v) and the quadrant of θ.
2. Relate to x, y, r:
   • If sin(θ) = v = y/r, we can assume r=1, so y=v, and x=±√(1-v²).
   • If cos(θ) = v = x/r, we can assume r=1, so x=v, and y=±√(1-v²).
   • If tan(θ) = v = y/x, we can assume x=1, y=v (or x=-1, y=-v, etc.), and r=√(x²+y²).
   • Similar logic applies for csc, sec, cot.
3. Determine Signs: The quadrant tells us the signs of x and y:
   • Quadrant I: x > 0, y > 0
   • Quadrant II: x < 0, y > 0
   • Quadrant III: x < 0, y < 0
   • Quadrant IV: x > 0, y < 0
   We use this to pick the correct sign for x or y found in step 2.
4. Calculate x, y, r: Based on the known function and quadrant, we find the specific values of x, y, and r (or their ratios, keeping r positive).
5. Calculate Other Functions:
   • sin(θ) = y/r
   • cos(θ) = x/r
   • tan(θ) = y/x
   • csc(θ) = r/y (undefined if y=0)
   • sec(θ) = r/x (undefined if x=0)
   • cot(θ) = x/y (undefined if y=0)

Variables Table

Variable	Meaning	Unit	Typical Range
v	Value of the known trigonometric function	Dimensionless	-1 ≤ v ≤ 1 for sin/cos, |v| ≥ 1 for csc/sec, any real for tan/cot
θ	The angle	Degrees or Radians	Any real number (often 0-360° or 0-2π rad)
x, y	Coordinates of a point on the terminal side	Length units (ratio matters)	Any real number
r	Distance from origin to (x,y)	Length units (ratio matters)	r > 0
Quadrant	Location of the terminal side of θ	I, II, III, or IV	1, 2, 3, or 4

Practical Examples (Real-World Use Cases)

Example 1: Given sin(θ)

Suppose you know sin(θ) = 3/5 and θ is in Quadrant II.
Input: Known Function = sin(θ), Value = 0.6, Quadrant = II.
Since sin(θ) = y/r = 3/5, we can take y=3, r=5.
x² + y² = r² => x² + 9 = 25 => x² = 16 => x = ±4.
In Quadrant II, x is negative, so x = -4.
We have x=-4, y=3, r=5.
Outputs:
sin(θ) = 3/5 = 0.6
cos(θ) = -4/5 = -0.8
tan(θ) = 3/-4 = -0.75
csc(θ) = 5/3 ≈ 1.6667
sec(θ) = 5/-4 = -1.25
cot(θ) = -4/3 ≈ -1.3333

Example 2: Given tan(θ)

Suppose you know tan(θ) = -1 and θ is in Quadrant IV.
Input: Known Function = tan(θ), Value = -1, Quadrant = IV.
Since tan(θ) = y/x = -1. In Quadrant IV, x > 0 and y < 0. We can take x=1, y=-1.
r = √(x² + y²) = √(1² + (-1)²) = √2 ≈ 1.4142.
We have x=1, y=-1, r=√2.
Outputs:
sin(θ) = -1/√2 ≈ -0.7071
cos(θ) = 1/√2 ≈ 0.7071
tan(θ) = -1
csc(θ) = -√2 ≈ -1.4142
sec(θ) = √2 ≈ 1.4142
cot(θ) = -1

For more basics, check our right triangle calculator.

How to Use This Find the Other Five Trigonometric Functions of θ Calculator

1. Select the Known Function: Choose the trigonometric function (sin, cos, tan, csc, sec, or cot) whose value you know from the "Known Trigonometric Function" dropdown.
2. Enter the Known Value: Type the numerical value of the selected function into the "Value of the Known Function" field. Ensure the value is within the valid range for that function (e.g., between -1 and 1 for sin and cos).
3. Select the Quadrant: Choose the quadrant (I, II, III, or IV) in which the angle θ lies from the "Quadrant of θ" dropdown. This is crucial for determining the signs of the other functions.
4. Read the Results: The calculator will automatically update and display the values of the other five trigonometric functions, along with the values of x, y, and r used in the calculation, in the "Results" section, the table, and the chart. The primary result highlights the calculated values.
5. Interpret: The table and chart provide a clear summary of all six function values.
6. Reset (Optional): Click "Reset" to clear the inputs and results to their default values for a new calculation.
7. Copy Results (Optional): Click "Copy Results" to copy the main results and intermediate values to your clipboard.

Understanding the unit circle can help visualize these relationships.

Key Factors That Affect Find the Other Five Trigonometric Functions of θ Results

1. Value of the Known Function: This is the starting point. An incorrect or out-of-range value will lead to errors or no solution (e.g., sin(θ) cannot be 2).
2. The Known Function Itself: Whether you start with sin, cos, tan, etc., determines the initial ratio (y/r, x/r, y/x, etc.) you work with.
3. Quadrant of θ: This is critical as it dictates the signs (+ or -) of x and y, and consequently the signs of the other trigonometric functions. The same function value in different quadrants yields different results for other functions.
4. Pythagorean Identity (x²+y²=r²): This fundamental relationship is used to find the third component (x, y, or r) once two are deduced from the known function value (assuming r=1 or x=1 or y=1 initially for ratios).
5. Reciprocal Identities: csc(θ)=1/sin(θ), sec(θ)=1/cos(θ), cot(θ)=1/tan(θ). These are used directly once sin, cos, and tan are found.
6. Quotient Identities: tan(θ)=sin(θ)/cos(θ), cot(θ)=cos(θ)/sin(θ). Also used in calculation and verification.

For a deeper dive into formulas, see our trigonometry formulas page.

Frequently Asked Questions (FAQ)

1. What if the given value for sin(θ) or cos(θ) is greater than 1 or less than -1?

The calculator will indicate an error or produce NaN (Not a Number) because the sine and cosine functions have a range of [-1, 1]. No real angle θ has sin(θ) or cos(θ) outside this range.

2. What if the given value for csc(θ) or sec(θ) is between -1 and 1?

Similarly, csc(θ) and sec(θ) values must be |v| ≥ 1. Values between -1 and 1 (exclusive) are invalid for these functions for real angles.

3. What if the known function is tan(θ) or cot(θ) and the value is 0?

If tan(θ) = 0, then sin(θ) = 0 and cos(θ) = ±1 (depending on quadrant, 0°/180°/360°). csc(θ) would be undefined, cot(θ) would be undefined if approached from specific angles but is generally considered undefined where tan=0 implies sin=0. Wait, if tan=0, sin=0, so csc is undef, cot is 1/0=undef. Oh, if tan=0, y=0, x!=0, so cot=x/0 undefined. If cot=0, x=0, y!=0, tan=y/0 undefined.

4. Why is the quadrant so important?

The quadrant determines the signs of x and y. For example, if sin(θ) = 0.5, θ could be in Quadrant I (30°) where cos(θ) is positive, or Quadrant II (150°) where cos(θ) is negative. Knowing the quadrant resolves this ambiguity.

5. Can I use this find the other five trigonometric functions of θ calculator for angles in radians?

Yes, the quadrant information is independent of whether the angle is measured in degrees or radians. The calculator works with the ratio values, not the angle measure itself directly.

6. What does it mean if a function value is "undefined"?

This occurs when the denominator in the x, y, r ratio is zero. For example, tan(90°) = y/x = 1/0, which is undefined. Similarly, csc(0°) = r/y = 1/0 is undefined.

7. How are x, y, and r determined?

We start by assuming one of them is 1 (or based on the given ratio) and use the given function value to find another. Then, x²+y²=r² finds the third. The quadrant gives the signs. For instance, if sin(θ)=0.5 (Q1), y/r=0.5, let r=1, y=0.5, x=√(1-0.25)=√0.75.

8. Does the calculator find the angle θ itself?

No, this calculator finds the values of the other five trigonometric functions of θ. To find θ itself, you would use inverse trigonometric functions (like arcsin, arccos, arctan), considering the quadrant. See our inverse trig functions calculator.

Related Tools and Internal Resources

• Right Triangle Calculator: Solves for sides and angles of a right triangle.
• Unit Circle Calculator: Visualizes angles and trigonometric function values on the unit circle.
• Trigonometry Formulas: A list of important trigonometric identities and formulas.
• Angle Conversion: Converts between degrees and radians.
• Pythagorean Theorem Calculator: Calculates a side of a right triangle given the other two.
• Inverse Trig Functions Calculator: Finds the angle given a trigonometric function value.