Find The Other Trigonometric Functions Calculator

Enter the value of one trigonometric function and the quadrant to find the values of the other five functions using this find the other trigonometric functions calculator.

Results Calculated All 6 trigonometric function values are shown below.

sin(θ):

cos(θ):

tan(θ):

csc(θ):

sec(θ):

cot(θ):

Angle (θ) ≈

Angle (θ) ≈

We use fundamental trigonometric identities like sin²θ + cos²θ = 1, tanθ = sinθ/cosθ, and reciprocal identities, adjusting signs based on the selected quadrant.

Summary of Trigonometric Function Values and Signs

Function	Value	Sign in Quadrant

What is a Find the Other Trigonometric Functions Calculator?

A find the other trigonometric functions calculator is a tool that helps you determine the values of all six trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) for a given angle, provided you know the value of at least one of these functions and the quadrant in which the angle lies. This is particularly useful when you have partial information about an angle and need to deduce the rest using fundamental trigonometric identities.

This calculator relies on identities such as sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ, and reciprocal identities (cscθ = 1/sinθ, secθ = 1/cosθ, cotθ = 1/tanθ). The quadrant information is crucial for determining the correct signs (+ or -) of the calculated functions, as each function is positive in two quadrants and negative in the other two.

Who Should Use It?

Students of trigonometry, algebra, calculus, and physics often use such a calculator. Engineers, scientists, and mathematicians also find it helpful when working with angles and their trigonometric relationships. Anyone needing to find all trig function values from limited information can benefit from this find the other trigonometric functions calculator.

Common Misconceptions

A common misconception is that knowing just one function's value is enough. However, without knowing the quadrant (or the sign of another function), the signs of the other functions remain ambiguous. For instance, if sinθ = 0.5, θ could be in Quadrant 1 (30°) or Quadrant 2 (150°), leading to different signs for cosθ and tanθ. Our find the other trigonometric functions calculator requires the quadrant to resolve this ambiguity.

Find the Other Trigonometric Functions Calculator: Formula and Mathematical Explanation

The find the other trigonometric functions calculator uses fundamental trigonometric identities to find the unknown values. The core identity is:

sin²θ + cos²θ = 1

From this, we can derive other relationships:

• If sin(θ) is known: |cos(θ)| = √(1 – sin²θ)
• If cos(θ) is known: |sin(θ)| = √(1 – cos²θ)

Other key identities include:

• tan(θ) = sin(θ) / cos(θ)
• csc(θ) = 1 / sin(θ)
• sec(θ) = 1 / cos(θ)
• cot(θ) = 1 / tan(θ) = cos(θ) / sin(θ)
• 1 + tan²(θ) = sec²(θ)
• 1 + cot²(θ) = csc²(θ)

The quadrant determines the sign of each function:

• Quadrant 1 (0° to 90°): All functions (sin, cos, tan, csc, sec, cot) are positive.
• Quadrant 2 (90° to 180°): sin and csc are positive; cos, tan, sec, cot are negative.
• Quadrant 3 (180° to 270°): tan and cot are positive; sin, cos, csc, sec are negative.
• Quadrant 4 (270° to 360°): cos and sec are positive; sin, tan, csc, cot are negative.

Variables Table

Variable	Meaning	Unit	Typical Range
sin(θ)	Sine of angle θ	Dimensionless	-1 to 1
cos(θ)	Cosine of angle θ	Dimensionless	-1 to 1
tan(θ)	Tangent of angle θ	Dimensionless	-∞ to ∞
csc(θ)	Cosecant of angle θ	Dimensionless	(-∞, -1] U [1, ∞)
sec(θ)	Secant of angle θ	Dimensionless	(-∞, -1] U [1, ∞)
cot(θ)	Cotangent of angle θ	Dimensionless	-∞ to ∞
θ	The angle	Degrees or Radians	0 to 360° or 0 to 2π rad
Quadrant	Location of the angle's terminal side	1, 2, 3, or 4	1, 2, 3, 4

Variables used in the find the other trigonometric functions calculator.

Practical Examples (Real-World Use Cases)

Example 1: Given sin(θ) and Quadrant

Suppose you know sin(θ) = 0.6 and the angle θ is in Quadrant 2.

1. Given: sin(θ) = 0.6, Quadrant = 2.
2. Find cos(θ): cos²θ = 1 – sin²θ = 1 – (0.6)² = 1 – 0.36 = 0.64. So, |cos(θ)| = √0.64 = 0.8. In Quadrant 2, cos is negative, so cos(θ) = -0.8.
3. Find tan(θ): tan(θ) = sin(θ) / cos(θ) = 0.6 / -0.8 = -0.75.
4. Find csc(θ): csc(θ) = 1 / sin(θ) = 1 / 0.6 = 1.6667.
5. Find sec(θ): sec(θ) = 1 / cos(θ) = 1 / -0.8 = -1.25.
6. Find cot(θ): cot(θ) = 1 / tan(θ) = 1 / -0.75 = -1.3333.

The find the other trigonometric functions calculator would give these values.

Example 2: Given tan(θ) and Quadrant

Suppose you know tan(θ) = -1.5 and the angle θ is in Quadrant 4.

1. Given: tan(θ) = -1.5, Quadrant = 4.
2. Find sec(θ): sec²θ = 1 + tan²θ = 1 + (-1.5)² = 1 + 2.25 = 3.25. So, |sec(θ)| = √3.25 ≈ 1.8028. In Quadrant 4, sec is positive, so sec(θ) ≈ 1.8028.
3. Find cos(θ): cos(θ) = 1 / sec(θ) = 1 / 1.8028 ≈ 0.5547.
4. Find sin(θ): sin(θ) = tan(θ) * cos(θ) = -1.5 * 0.5547 ≈ -0.8321.
5. Find csc(θ): csc(θ) = 1 / sin(θ) = 1 / -0.8321 ≈ -1.2018.
6. Find cot(θ): cot(θ) = 1 / tan(θ) = 1 / -1.5 ≈ -0.6667.

Our find the other trigonometric functions calculator quickly provides these results.

How to Use This Find the Other Trigonometric Functions Calculator

1. Select Known Function: Choose the trigonometric function (sin, cos, tan, csc, sec, or cot) whose value you know from the "Known Function" dropdown.
2. Enter Known Value: Input the numerical value of the selected function into the "Value of Known Function" field. Ensure the value is valid for the chosen function (e.g., between -1 and 1 for sin and cos).
3. Select Quadrant: Choose the quadrant (1, 2, 3, or 4) where the angle θ lies from the "Quadrant" dropdown.
4. Calculate: The calculator automatically updates as you enter valid inputs, or you can click "Calculate".
5. Read Results: The "Results" section will display the values of all six trigonometric functions (sin(θ), cos(θ), tan(θ), csc(θ), sec(θ), cot(θ)), the approximate angle in degrees and radians within the specified quadrant, a bar chart, and a summary table.
6. Reset: Click "Reset" to clear the inputs and results to their default values.
7. Copy: Click "Copy Results" to copy the main results and inputs to your clipboard.

Using the find the other trigonometric functions calculator is straightforward and gives immediate results based on fundamental identities.

Key Factors That Affect Find the Other Trigonometric Functions Calculator Results

• Value of the Known Function: The numerical value directly influences the magnitudes of the other functions through identities like sin²θ + cos²θ = 1. An invalid value (e.g., sin(θ) = 2) will yield no real solutions for other functions.
• Type of Known Function: Whether you know sin, cos, tan, etc., determines which identities are used first to find the others.
• Quadrant of the Angle: This is crucial for determining the signs (+ or -) of the calculated trigonometric functions. The same magnitude for a function can correspond to angles in different quadrants, but the signs of other functions will differ.
• Trigonometric Identities: The accuracy of the results depends on the correct application of fundamental identities (Pythagorean, reciprocal, quotient).
• Domain and Range of Functions: Input values must be within the valid range for the selected function (e.g., |sin(θ)| ≤ 1, |sec(θ)| ≥ 1). Values outside these ranges will result in errors or non-real answers. Our find the other trigonometric functions calculator validates these.
• Rounding: The precision of the results depends on the rounding applied during intermediate calculations, especially when dealing with square roots. Our calculator aims for reasonable precision.

Frequently Asked Questions (FAQ)

Q: What if the value I enter for sin(θ) or cos(θ) is greater than 1 or less than -1? A: The find the other trigonometric functions calculator will indicate an error because the sine and cosine functions only have values between -1 and 1, inclusive, for real angles.

Q: What if the value for sec(θ) or csc(θ) is between -1 and 1? A: Similarly, the calculator will show an error, as the secant and cosecant functions have values |sec(θ)| ≥ 1 and |csc(θ)| ≥ 1 for real angles.

Q: How does the calculator determine the angle θ? A: It calculates a reference angle using the inverse trigonometric function of the absolute value of the known function (e.g., arcsin(|value|)), and then adjusts it based on the specified quadrant to find the angle θ between 0° and 360°.

Q: Can I use this calculator if I only know the signs of two functions but not their values? A: No, this find the other trigonometric functions calculator requires the numerical value of at least one function and the quadrant (or enough information to determine it). Knowing only signs narrows down the quadrant but not the values.

Q: What if tan(θ) or cot(θ) is undefined? A: This occurs when cos(θ)=0 (for tan, sec) or sin(θ)=0 (for cot, csc). The calculator will show "Undefined" or handle it based on the input. For instance, if cos(θ)=0, tan(θ) is undefined.

Q: Why is the quadrant so important? A: The quadrant determines the signs of the trigonometric functions. For example, if you know sin(θ) = 0.5, cos(θ) could be √0.75 or -√0.75. The quadrant tells you which sign is correct.

Q: Does this calculator work with radians? A: It primarily uses degrees for angle input via quadrant but shows the resulting angle in both degrees and radians.

Q: Can I find the angle if I know two sides of a right triangle? A: Yes, you can find one trig function (like sin = opposite/hypotenuse) and then use this find the other trigonometric functions calculator if you also know the quadrant (usually Quadrant 1 for simple right triangles). Or use our right triangle solver.

Related Tools and Internal Resources

• Trigonometry Basics: Learn the fundamentals of trigonometric functions and identities.
• Unit Circle Calculator: Explore the unit circle and the values of trigonometric functions at different angles.
• Right Triangle Solver: Calculate angles and sides of a right triangle.
• Law of Sines Calculator: Solve non-right triangles using the Law of Sines.
• Law of Cosines Calculator: Solve non-right triangles using the Law of Cosines.
• Angle Conversion Tool: Convert between degrees, radians, and other angle units.