Problem 106 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate the function for the given values of \(t\). Values of \(t\) (a) \(t=3\) (a) \(t=2\) (a) \(t=0.3\) (a) \(t=0.1\) (b) \(t=\pi\) (b) \(t=0\) (b) \(t=-\pi / 4\) (b) \(t=-2 \pi / 3\) Trigonometric Function $$f(t)=\cos t$$

Step-by-Step Solution

Answer

The values of \( f(t) = \cos t \) at the provided points are: \( f(3) = -0.9900 \), \( f(2) = -0.4161 \), \( f(0.3) = 0.9553 \), \( f(0.1) = 0.9950 \), \( f(\pi) = -1 \), \( f(0) = 1 \), \( f(-\pi/4) = 0.7071 \), \( f(-2\pi/3) = -0.5000 \)

Step 1: Evaluate \( f(t) = \cos t \) for \( t = 3 \)

Replace \( t \) with 3 in the function. The value will be \( f(3) = \cos(3) \). Using a calculator the value will be approximately equal to -0.9900.

Step 2: Evaluate \( f(t) = \cos t \) for \( t = 2 \)

Replace \( t \) with 2 in the function. The value will be \( f(2) = \cos(2) \). Using a calculator the value will be approximately equal to -0.4161.

Step 3: Evaluate \( f(t) = \cos t \) for \( t = 0.3 \)

Replace \( t \) with 0.3 in the function. The value will be \( f(0.3) = \cos(0.3) \). Using a calculator, this will be approximately equal to 0.9553.

Step 4: Evaluate \( f(t) = \cos t \) for \( t = 0.1 \)

Replace \( t \) with 0.1 in the function. The value will be \( f(0.1) = \cos(0.1) \). Using a calculator, this will be approximately equal to 0.9950.

Step 5: Evaluate \( f(t) = \cos t \) for \( t = \pi \)

Replace \( t \) with \( \pi \) in the function. The value will be \( f(\pi) = \cos(\pi) \). We know that \(\cos(\pi) = -1\)

Step 6: Evaluate \( f(t) = \cos t \) for \( t = 0 \)

Replace \( t \) with 0 in the function. The value will be \( f(0) = \cos(0) \). We know that \(\cos(0) = 1 \)

Step 7: Evaluate \( f(t) = \cos t \) for \( t = -\pi / 4 \)

Replace \( t \) with \( -\pi/4 \) in the function. The value will be \( f(-\pi/4) = \cos(-\pi/4) \). Using a calculator, this will be approximately equal to 0.7071.

Step 8: Evaluate \( f(t) =\cos t \) for \( t = -2\pi/3 \)

Replace \( t \) with \( -2\pi/3 \) in the function. The value will be \( f(-2\pi/3) = \cos(-2\pi/3) \). Using a calculator, this will be approximately equal to -0.5000.

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos t \), is one of the fundamental trigonometric functions used to link angles to the ratios of sides in right-angled triangles. But its use extends beyond just triangles, describing wave patterns like sound and light. The cosine function provides the horizontal coordinate of a unit circle corresponding to a particular angle \( t \). This results in values that range between \(-1\) and \(1\).

• When \( t = 0 \), the cosine is \(1\).
• At \( t = \pi \), the cosine is \(-1\).
• For angles \( t \) like \( \pi/2 \) and \( 3\pi/2 \), the cosine takes the value \(0\).

Cosine is an even function, meaning that \( \cos(-t) = \cos(t) \). This symmetry makes it easier to determine values for negative angles.

Evaluation of Functions

Evaluating a function involves finding the value of \( f(t) \) for given inputs. For \( f(t) = \cos t \), we substitute \( t \) with the given values and use either trigonometric knowledge or a calculator to find \( \cos(t) \).

• For example, \( f(3) = \cos(3) \), which can be approximately calculated with a calculator.
• Standard angles like \( \pi \) or \( 0 \) have known cosine values: \(\cos(\pi) = -1 \) and \( \cos(0) = 1 \).

When using a calculator, ensure it's set to the correct mode (radians or degrees) depending on your input angle \( t \), as this affects the outcome of your evaluation.

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. It's a crucial tool in trigonometry for understanding angle and trigonometric function relationships.

• The cosine of an angle \( t \) in the unit circle corresponds to the x-coordinate of the point on the circle's circumference.
• This visualization helps bind the theoretical understanding of trigonometric functions to geometric representations.

The unit circle aids in converting between angle measures and finding precise trigonometric function values for angles, bridging algebraic and geometric aspects of trigonometry.

Angle Measurement

Angles in trigonometry can be measured in degrees or radians. For the cosine function, radians are commonly used because they seamlessly integrate with calculus and other branches of mathematics.

• One complete revolution around the circle is \( 360\) degrees or \( 2\pi \) radians.
• Understanding radians helps in simplifying the measurement of angles, especially those not marked by common degree values, like \( \pi/4 \) or \( -2\pi/3 \).

It's essential to become familiar with converting between degrees and radians: \( 180 \) degrees equals \( \pi \) radians, providing a standardized conversion factor for calculations.