Problem 35 - Algebra and Trigonometry

Question

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\) $$ \left(-\frac{20}{29}, \frac{21}{29}\right) $$

Step-by-Step Solution

Answer

\(\sin t = \frac{21}{29}, \cos t = -\frac{20}{29}, \tan t = -\frac{21}{20}\).

Step 1: Identify Components of P(x, y)

The terminal point is given as \( P\left( -\frac{20}{29}, \frac{21}{29} \right) \). This point \((x, y)\) represents \( \left( \cos t, \sin t \right) \) on the unit circle.

Step 2: Determine \(\sin t\) and \(\cos t\)

Since the terminal point \( P(x, y) \) is given by \( \left( \cos t, \sin t \right) \), we directly extract these values: \( \cos t = -\frac{20}{29} \) and \( \sin t = \frac{21}{29} \).

Step 3: Calculate \(\tan t\)

The tangent of the angle \( t \) can be found using the formula \( \tan t = \frac{y}{x} \). Substituting the values from \( P(x, y) \):\[tan t = \frac{\frac{21}{29}}{-\frac{20}{29}} = -\frac{21}{20}\]

Key Concepts

Unit Circle

The unit circle is a powerful concept in trigonometry, fundamentally linking angles and trigonometric functions. Picture a circle with a radius of one, centered at the origin of a coordinate plane. Every point on this circle represents the cosine and sine values of an angle.

When an angle is formed by rotating a ray around the circle, it intersects the circle at a point \( P(x, y) \). Here, \( x \) and \( y \) are the coordinates corresponding to \( \cos t \) and \( \sin t \) respectively.

• This circle helps in visualizing the trigonometric functions for any angle \( t \).
• A complete revolution around the circle covers \( 2\pi \) radians or 360 degrees.
• Angles can be positive (counter-clockwise) or negative (clockwise).

Understanding the unit circle is crucial as it provides a geometric framework for defining the sine, cosine, and tangent functions based on angle \( t \).

Sine Function

The sine function is a fundamental trigonometric function representing the vertical component of a point on the unit circle. For an angle \( t \), the sine function is defined as the \( y \)-coordinate of the point where the terminal side of the angle intersects the unit circle.

In mathematical terms, this is expressed as:
\[ \sin t = y \]
For the given point \( P\left( -\frac{20}{29}, \frac{21}{29} \right) \), \( \sin t \) is directly extracted as \( \frac{21}{29} \). This value represents how far above or below the x-axis the point lies.

• The sine function varies between -1 and 1 because it's a circle with a radius of 1.
• Sine of small angles approaches the angle itself in radian measure, illustrating close approximation for tiny angles.

Cosine Function

The cosine function represents the horizontal component of a point on the unit circle. The cosine of an angle \( t \) is simply the \( x \)-coordinate where the terminal side of that angle meets the unit circle.

Expressed mathematically, this is shown as:
\[\cos t = x \]
For our specific example with point \( P\left( -\frac{20}{29}, \frac{21}{29} \right) \), we find that \( \cos t = -\frac{20}{29} \). This value indicates how far left or right from the y-axis the point is positionally.

• The cosine values also oscillate between -1 and 1.
• When \( t = 0 \), \( \cos t \) reaches its maximum of 1.

The cosine function is vital for understanding motions and positions related to angles and plays a key role in the computation of curves and oscillations.

Tangent Function

The tangent function is involved with the ratio of the sine function to the cosine function. It provides insights into the slope of the line that connects the origin to a point \( P(x, y) \) on the unit circle.

The formula to determine \( \tan t \) is:\[tan t = \frac{\sin t}{\cos t} = \frac{y}{x}\]
In our case:\[\tan t = \frac{\frac{21}{29}}{-\frac{20}{29}} = -\frac{21}{20}\]
The tangent function can take any real value, and it is absent in angles where the cosine function becomes zero (i.e., at \( \frac{\pi}{2} \), \( \frac{3\pi}{2} \), etc.), as this leads to division by zero.

• The tangent is periodic with a period of \( \pi \), unlike sine and cosine which have a period of \( 2\pi \).
• Studying tangent helps in understanding slopes and gradients in trigonometric contexts.