Cosine Function in Math – Definition, Formula, Examples

The cosine function (cos) relates an angle of a right triangle to the ratio of the length of the adjacent side (the side next to the angle) to the hypotenuse (the longest side of the triangle). In the context of the unit circle (a circle with radius 1), the cosine of an angle corresponds to the x-coordinate of a point on the circle’s circumference.

Placing Cosine in Context

Cosine is one of the primary trigonometric functions. The other key functions include:

• Sine (sin): Represents the ratio of the length of the side opposite an angle to the hypotenuse in a right triangle. On the unit circle, it gives the y-coordinate.
• Tangent (tan): Defined as the ratio of sine to cosine, or the ratio of the opposite side to the adjacent side in a right triangle.

Together, these functions allow us to analyze and solve problems involving angles and lengths in various branches of mathematics and science.

Key Takeaways: Cosine Function

• Definition: Cosine is the ratio of the adjacent side to the hypotenuse in a right triangle, or the x-coordinate of a point on the unit circle.
• Range: The values of cosine lie between –1 and 1.
• Periodicity: Cosine is a periodic function with a period of 360° (or 2π radians), meaning it repeats every full circle.
• Even Function: Cosine is even, so cos⁡(−θ) = cos⁡(θ).
• Graph Shape: The cosine graph is a smooth, continuous wave that starts at its maximum value when the angle is 0°.
• Applications: Used in geometry (e.g., law of cosines), calculus (derivatives and integrals), physics (modeling oscillations), and engineering.

Cosine Definition

Cosine in a Right Triangle

For a right triangle with an angle θ:

cos⁡(θ) = Adjacent Side / Hypotenuse

• Adjacent Side: The side that forms the angle θ along with the hypotenuse.
• Hypotenuse: The longest side of the right triangle, opposite the right angle.

Cosine on the Unit Circle

In the unit circle (a circle with radius 1), any angle θ in standard position corresponds to a point (x,y) on the circle. The cosine of the angle is the x-coordinate:

cos⁡(θ) = x

Cosine Formula and Worked Example

Formula for Cosine (Right Triangle Definition)

cos⁡(θ) = Adjacent Side / Hypotenuse

Worked Example Calculation

Problem:
Given a right triangle where the length of the adjacent side to angle θ is 4 and the hypotenuse is 5, find cos⁡(θ).

Solution:

cos⁡(θ) = 4 / 5 = 0.8

So, the cosine of the angle θ is 0.8.

Cosine Table

Below is a table showing cosine values for common angles (in degrees):

Cosine Properties in Different Quadrants

Note that the sign (positive or negative) of the cosine function changes depending on the quadrant of the angle in standard position.

Quadrant Table

The Cosine Graph

Description of the Cosine Graph

• Shape: The cosine graph is a smooth, continuous wave that oscillates between 1 and -1.
• Period: It has a period of 360° (or 2π radians); the pattern repeats every full rotation.
• Amplitude: The amplitude is 1, meaning the maximum displacement from the center line (0) is 1.
• Phase Shift: The cosine graph starts at its maximum value (1) when the angle is 0°.

Comparison with the Sine Graph

• The sine graph starts at 0 when the angle is 0° and reaches its maximum at 90°.
• The cosine graph is essentially a sine graph shifted to the left by 90° (or π/2​ radians). In other words, cos⁡(θ) = sin⁡(θ + 90°).

Key Cosine Identities

There are several important identities that involve the cosine function:

1. Pythagorean Identity: sin⁡²(θ) + cos⁡²(θ) = 1
2. Cosine of a Negative Angle: cos⁡(−θ) = cos⁡(θ)
3. Cosine Sum Formula: cos⁡(α + β) = cos⁡(α)cos⁡(β) − sin⁡(α)sin⁡(β)
4. Cosine Difference Formula: cos⁡(α − β) = cos⁡(α)cos⁡(β) + sin⁡(α)sin⁡(β)
5. Double-Angle Formula: cos⁡(2θ) = 2cos^⁡2(θ) − 1
   Alternatively:
   cos⁡(2θ) = 1 − 2sin⁡²(θ)

The Law of Cosines

The Law of Cosines is a generalization of the Pythagorean theorem for any triangle (not just right triangles). It relates the lengths of the sides of a triangle to the cosine of one of its angles.

For a triangle with sides a, b, and c and an angle C opposite side c, the law is given by:

c² = a² + b² − 2abcos⁡(C)

This formula is particularly useful for:

• Finding the third side of a triangle when two sides and the included angle are known.
• Determining the angles of a triangle when all three sides are known.

Cosine in Calculus

Cosine functions play an important role in calculus.

Derivative of Cosine

d/dx[cos⁡(x)] = −sin⁡(x)

Integral of Cosine

∫cos⁡(x) dx = sin⁡(x) + C

where C is the constant of integration.

Worked Math Problems Involving Cosine

Here are some sample problems that students might encounter in geometry, trigonometry, or calculus courses:

1. Right Triangle Problem:
   • Given: In a right triangle, the adjacent side to angle θ is 7, and the hypotenuse is 10.
   • Task: Calculate cos⁡(θ).
2. Unit Circle Problem:
   • Given: An angle θ = 150°.
   • Task: Determine cos⁡(150°) using the unit circle and explain why it is negative.
3. Using the Law of Cosines:
   • Given: In triangle ABC, sides a=8, b=6, and angle C=60°.
   • Task: Use the law of cosines to find the length of side c.
4. Graph Analysis:
   • Task: Sketch the graphs of y = cos⁡(x) and y = sin⁡(x) on the same axes, and describe their phase relationship.
5. Cosine Identity Verification:
   • Task: Prove the identity cos⁡(2θ) = 2cos^⁡2(θ) − 1 using the double-angle formulas.
6. Calculus Application:
   • Task: Find the derivative of f(x) = cos⁡(3x) and then integrate f(x)=cos⁡(3x) with respect to x.
7. Real-World Application:
   • Task: A pendulum swings such that its displacement from the center is modeled by d(t) = 5cos⁡(2t). Determine the period and amplitude of the pendulum’s motion.

The answers to these problems are at the bottom of the article.

Cosine in Complex Numbers and Euler’s Formula

One of the most elegant connections in mathematics is found in Euler’s formula, which links complex numbers with trigonometric functions. Euler’s formula states that for any real number θ:

e^iθ = cos⁡(θ) + i sin⁡(θ)

where:

• e is the base of the natural logarithm,
• i is the imaginary unit (with i² = −1),
• cos⁡(θ) and sin⁡(θ) are the familiar cosine and sine functions, respectively.

Key Points:

• Interrelation:
  Euler’s formula shows that exponential functions with imaginary exponents naturally encode both cosine and sine, unifying these trigonometric functions with complex exponential functions.
• Cosine Extraction:
  By taking the real part of Euler’s formula, we see that:
  cos⁡(θ) = ℜ(e^iθ)
• Implications in Mathematics:
  This relationship is pivotal in many areas such as Fourier analysis, where complex exponentials are used to represent periodic functions. It also provides insights into the behavior of trigonometric functions and helps simplify the computation of integrals and derivatives in calculus.

History

The study of trigonometric functions, including cosine, has a rich history that spans many centuries and cultures:

• Ancient Beginnings:
  Early trigonometric ideas traced back to ancient civilizations such as the Babylonians and Egyptians, who used simple geometric principles to solve problems related to astronomy and land measurement.
• Greek Contributions:
  The ancient Greeks, notably mathematicians like Hipparchus and Ptolemy, played a pivotal role in developing trigonometry. Ptolemy’s work in his seminal text Almagest involved extensive trigonometric tables based on chords in a circle—a precursor to our modern sine and cosine functions.
• Indian and Islamic Advances:
  Indian mathematicians further refined trigonometric methods, introducing sine (known as “jya”) and its related functions. Later, Islamic scholars translated and expanded these works, developing comprehensive trigonometric tables that were later transmitted to Europe.
• Renaissance to Modern Times:
  With the advent of the Renaissance, European mathematicians built upon earlier works. The introduction of the unit circle and the formal definition of trigonometric functions, including cosine, provided the foundation for modern trigonometry. Today, cosine is integral to various fields such as physics, engineering, and computer science.

Common Mistakes and Misconceptions

When working with the cosine function, students and practitioners alike may encounter several pitfalls. Here are some of the most common mistakes along with tips to avoid them:

• Mixing Degrees and Radians:
  • Mistake: Using degrees in formulas or computations that require radians (or vice versa).
  • Tip: Always verify the units of your angle measurements before plugging them into formulas. Remember that 360° = 2π radians.
• Sign Errors in Different Quadrants:
  • Mistake: Forgetting that cosine is positive in the first and fourth quadrants and negative in the second and third quadrants.
  • Tip: Use the mnemonic “All Students Take Calculus” (ASTC) to remember the sign conventions for trigonometric functions in each quadrant.
• Overlooking the Even Nature of Cosine:
  • Mistake: Incorrectly assuming cos⁡(−θ) ≠ cos⁡(θ).
  • Tip: Remember that cosine is an even function, so cos⁡(−θ) = cos⁡(θ).
• Misapplying Formulas:
  • Mistake: Confusing the cosine double-angle formula with other trigonometric identities.
  • Tip: Write down the relevant identities and ensure you are using the correct formula for the given problem.
• Ignoring Graphical Interpretations:
  • Mistake: Focusing solely on numerical or algebraic methods without understanding the graphical behavior of the cosine function.
  • Tip: Graphing the function can provide valuable insights into its periodicity, amplitude, and phase shifts, reinforcing conceptual understanding.

Being mindful of these common errors can lead to greater accuracy and confidence when working with cosine.

Frequently Asked Questions (FAQs)

Q1: Why is the cosine function defined as the x-coordinate on the unit circle?

A: In the unit circle, each angle θ corresponds to a point (x,y). The cosine of θ is defined as the x-coordinate because it represents the horizontal distance from the origin to the point on the circle, thereby naturally relating an angle to a ratio that is consistent for all circles of radius 1.

Q2: How do I convert between degrees and radians?

A: The conversion is based on the equivalence:

360° = 2π radians or 1° = π/180 radians

To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π​.

Q3: Why is cosine considered an even function?

A: A function is even if it satisfies f(−x) = f(x). Since the cosine function meets this criterion (cos⁡(−θ) = cos⁡(θ)), it is an even function. This symmetry about the vertical axis (y-axis) is a fundamental property of the cosine graph.

Q4: What are some common applications of the cosine function?

A: The cosine function finds use in various fields:

• Geometry: For solving triangles via the Law of Cosines.
• Calculus: For differentiation and integration.
• Physics: In modeling oscillations, waves, and harmonic motion.
• Engineering and Signal Processing: In Fourier analysis and to measure cosine similarity in data science.

Q5: What is the relationship between cosine and sine?

A: Cosine and sine are closely related functions. One important relationship is the phase shift:

cos⁡(θ) = sin⁡(θ + π/2)

This means the cosine function is a sine function shifted to the left by π/2​ radians (or 90°).

References

• Abramowitz, Milton; Stegun, Irene A. (1970). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover Publications.
• Axler, Sheldon (2012). Algebra and Trigonometry. John Wiley & Sons. ISBN 978-0470-58579-5.
• Howie, John M. (2003). Complex Analysis. Springer Undergraduate Mathematics Series. Springer. doi:10.1007/978-1-4471-0027-0. ISBN 978-1-4471-0027-0.
• Merlet, Jean-Pierre (2004). “A Note on the History of the Trigonometric Functions” in Ceccarelli, Marco (ed.). International Symposium on History of Machines and Mechanisms. Springer. doi:10.1007/1-4020-2204-2. ISBN 978-1-4020-2203-6.

Problem 1: Right Triangle Problem

Given: In a right triangle, the adjacent side to angle θ is 7, and the hypotenuse is 10.

Task: Calculate cos⁡(θ).

Solution:

The definition of cosine in a right triangle is:

cos⁡(θ) = Adjacent Side / Hypotenuse

Plug in the given values:

cos⁡(θ) = 7 / 10 = 0.7

Answer: cos⁡(θ) = 0.7

Problem 2: Unit Circle Problem

Given: θ=150°.

Task: Determine cos⁡(150°) using the unit circle and explain why it is negative.

Solution:

1. Determine the Reference Angle:
   • The reference angle for 150° is: 180°−150° = 30°
2. Find the Cosine of the Reference Angle:
   • We know that: cos⁡(30°) = √3/2 ≈ 0.866
3. Determine the Sign:
   • 150° lies in the second quadrant. In the unit circle, cosine (the x-coordinate) is negative in the second quadrant.
4. Apply the Sign: cos⁡(150°) = −cos⁡(30°) = −√3/2 ≈ −0.866

Answer: cos⁡(150°) ≈ −0.866

Problem 3: Using the Law of Cosines

Given: In triangle ABC, sides a=8, b=6, and angle C=60°.

Task: Use the law of cosines to find the length of side c.

Solution:

The Law of Cosines is given by: c² = a² + b² − 2ab cos⁡(C)

1. Plug in the values: c² = 8² + 6² − 2⋅8⋅6⋅cos⁡(60°)
2. Calculate individual components:
   • 8² = 64
   • 6² = 36
   • Sum: 64 + 36 = 100
   • 2⋅8⋅6 = 96
   • cos⁡(60°) = 0.5
3. Compute: c² = 100 − 96⋅0.5 = 100 − 48 = 52
   c = √52 =2√13 ≈ 7.21

Answer: c ≈ 7.21

Problem 4: Graph Analysis

Task: Sketch the graphs of y = cos⁡(x) and y = sin⁡(x) on the same axes, and describe their phase relationship.

Solution:

While a hand-drawn graph is not possible here, we can describe the key features:

• Graph of y = cos⁡(x):
  • Starting Point: y = cos⁡(x) starts at 1 when x=0.
  • Key Points:
    • x = π/2​: cos⁡(π/2) = 0
    • x = π: cos⁡(π) = −1
    • x = 3π/2​: cos⁡(3π/2) = 0
    • x = 2π: cos⁡(2π) = 1
• Graph of y = sin⁡(x):
  • Starting Point: y = sin⁡(x) starts at 0 when x=0.
  • Key Points:
    • x = π/2​: sin⁡(π/2) = 1
    • x =π: sin⁡(π) = 0
    • x = 3π/2​: sin⁡(3π/2) = −1
    • x = 2π: sin⁡(2π) = 0
• Phase Relationship:
  • The cosine graph is essentially the sine graph shifted to the left by π/2​ radians:
    cos⁡(x) = sin⁡(x + π/2)
  • This means that the peaks, zeros, and troughs of the cosine function occur π/2​ radians earlier than those of the sine function.

Answer:

• Cosine Graph: Starts at 1 (maximum) at x = 0, crosses zero at x = π/2​ and x = 3π/2​, and reaches −1 at x = π.
• Sine Graph: Starts at 0 at x=0, reaches 1 at x =π/2​, crosses zero at x = π, and reaches −1 at x = 3π/2.
• Phase Shift: cos⁡(x) is the same as sin⁡(x + π/2).

Problem 5: Cosine Identity Verification

Task: Prove the identity: cos⁡(2θ) = 2cos⁡²(θ) − 1

Solution:

1. Start with the double-angle formula for cosine: cos⁡(2θ) = cos⁡²(θ) − sin⁡²(θ)
2. Use the Pythagorean identity: sin⁡²(θ) = 1−cos⁡²(θ)
3. Substitute into the double-angle formula: cos⁡(2θ) = cos^⁡2(θ) − (1− cos^⁡2(θ)) = cos^⁡2(θ) − 1 + cos^⁡2(θ)
4. Combine like terms: cos⁡(2θ) = 2 cos^⁡2(θ) − 1

Answer: The identity is verified: cos⁡(2θ) = 2cos^⁡2(θ)−1.

Problem 6: Calculus Application

Task: Find the derivative of f(x) = cos⁡(3x) and then integrate f(x) = cos⁡(3x) with respect to x.

Solution:

1. Derivative

• Use the chain rule: d/dx[cos⁡(3x)] = −sin⁡(3x) ⋅ d/dx[3x]
• Differentiate 3x: d/dx[3x] = 3
• Thus, f′(x) = −3sin⁡(3x)

2. Integral

• Setup the integral: ∫cos⁡(3x) dx
• Let u=3x; then du = 3dx or dx=du / 3​.
• Substitute: ∫cos⁡(u) du/3 = 1/3 ∫cos⁡(u) du
• Integrate: ∫cos⁡(u) du = sin⁡(u) + C
• Substitute back u=3x: ∫cos⁡(3x) dx = 1/3 sin⁡(3x) + C

Answer:

• Derivative: f′(x)=−3sin⁡(3x)
• Integral: ∫cos⁡(3x) dx = 1/3 sin⁡(3x)+C

Problem 7: Real-World Application (Pendulum Motion)

Given: The displacement of a pendulum is modeled by d(t) = 5cos⁡(2t)

Task: Determine the amplitude and period of the pendulum’s motion.

Solution:

1. Amplitude:
   The amplitude is the coefficient of the cosine function. Amplitude =5
2. Period:
   For a function of the form A cos⁡(Bt), the period T is: T = 2π / ∣B∣
   ​Here, B=2, so: T = 2π / 2 = π

Answer:

• Amplitude: 5 units
• Period: π time units