Finding Exact Values Of Trigonometric Functions For Point (-4, 7)

by THE IDEN 66 views

In trigonometry, determining the exact values of trigonometric functions is a fundamental skill. When given a point on the terminal side of an angle in standard position, we can use the definitions of trigonometric functions and the Pythagorean theorem to find these values. This article will walk you through the process step-by-step, ensuring you grasp the underlying concepts and can apply them to various problems.

Understanding Standard Position and Trigonometric Functions

Before diving into the specifics of our problem, let's clarify some key concepts. An angle is in standard position when its vertex is at the origin (0,0) of the coordinate plane, and its initial side lies along the positive x-axis. The terminal side is the ray that rotates from the initial side, and the angle $ heta$ is the measure of this rotation.

The six trigonometric functions—sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot)—relate the angles of a right triangle to the ratios of its sides. When dealing with angles in standard position, we can define these functions in terms of the coordinates (x, y) of a point on the terminal side and the distance r from the origin to that point. The definitions are as follows:

  • Sine (sinθ{sin \theta}): The ratio of the y-coordinate to the distance r. Expressed as: sinθ=yr{sin \theta = \frac{y}{r}}
  • Cosine (cosθ{cos \theta}): The ratio of the x-coordinate to the distance r. Expressed as: cosθ=xr{cos \theta = \frac{x}{r}}
  • Tangent (tanθ{tan \theta}): The ratio of the y-coordinate to the x-coordinate. Expressed as: tanθ=yx{tan \theta = \frac{y}{x}}
  • Cosecant (cscθ{csc \theta}): The reciprocal of sine, the ratio of the distance r to the y-coordinate. Expressed as: cscθ=ry{csc \theta = \frac{r}{y}}
  • Secant (secθ{sec \theta}): The reciprocal of cosine, the ratio of the distance r to the x-coordinate. Expressed as: secθ=rx{sec \theta = \frac{r}{x}}
  • Cotangent (cotθ{cot \theta}): The reciprocal of tangent, the ratio of the x-coordinate to the y-coordinate. Expressed as: cotθ=xy{cot \theta = \frac{x}{y}}

Finding r Using the Pythagorean Theorem

The distance r from the origin to the point (x, y) is crucial for calculating the trigonometric functions. We can find r using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In our context, r is the hypotenuse, and x and y are the other two sides. Thus, we have: r=x2+y2{r = \sqrt{x^2 + y^2}}

This formula allows us to determine r when given the coordinates (x, y) of a point on the terminal side.

Step-by-Step Solution for the Point (-4, 7)

Now, let's apply these concepts to the specific problem at hand: finding the exact values of the six trigonometric functions for an angle $ heta$ whose terminal side passes through the point (-4, 7). This section will guide you through each step of the calculation, ensuring a clear understanding of the process.

Step 1: Identify x and y

The given point is (-4, 7), so we can identify the coordinates as:

  • x = -4
  • y = 7

These values are the foundation for our calculations, representing the horizontal and vertical components, respectively, of the point on the terminal side.

Step 2: Calculate r

Using the Pythagorean theorem, we can find the distance r from the origin to the point (-4, 7): r=x2+y2=(−4)2+72=16+49=65{r = \sqrt{x^2 + y^2} = \sqrt{(-4)^2 + 7^2} = \sqrt{16 + 49} = \sqrt{65}}

Thus, r is equal to $\sqrt{65}$. This value is essential for determining the trigonometric function values, as it serves as the denominator in sine and cosine and appears in the other functions as well.

Step 3: Calculate the Six Trigonometric Functions

Now that we have the values of x, y, and r, we can calculate the six trigonometric functions using their definitions:

  1. Sine (sin θ): sinθ=yr=765{sin \theta = \frac{y}{r} = \frac{7}{\sqrt{65}}} To rationalize the denominator, we multiply both the numerator and denominator by $\sqrt65}$ ${sin \theta = \frac{7\sqrt{65}{65}}$

The sine of $\theta$ is the ratio of the y-coordinate to the distance r. In this case, the sine is positive, which aligns with the fact that the point (-4, 7) lies in the second quadrant where the sine function is positive.

  1. Cosine (cos θ): cosθ=xr=−465{cos \theta = \frac{x}{r} = \frac{-4}{\sqrt{65}}} Rationalizing the denominator: cosθ=−46565{cos \theta = \frac{-4\sqrt{65}}{65}}

Cosine represents the ratio of the x-coordinate to the distance r. Here, the cosine value is negative, consistent with the x-coordinate being negative in the second quadrant.

  1. Tangent (tan θ): tanθ=yx=7−4=−74{tan \theta = \frac{y}{x} = \frac{7}{-4} = -\frac{7}{4}}

The tangent is the ratio of the y-coordinate to the x-coordinate. The negative value indicates that the angle is in a quadrant where tangent is negative, which is the second quadrant in this instance.

  1. Cosecant (csc θ): cscθ=ry=657{csc \theta = \frac{r}{y} = \frac{\sqrt{65}}{7}}

Cosecant is the reciprocal of sine. It's the ratio of the distance r to the y-coordinate. The value is positive, as expected in the second quadrant.

  1. Secant (sec θ): secθ=rx=65−4=−654{sec \theta = \frac{r}{x} = \frac{\sqrt{65}}{-4} = -\frac{\sqrt{65}}{4}}

Secant is the reciprocal of cosine, the ratio of r to the x-coordinate. The negative sign aligns with the cosine's negative value in the second quadrant.

  1. Cotangent (cot θ): cotθ=xy=−47=−47{cot \theta = \frac{x}{y} = \frac{-4}{7} = -\frac{4}{7}}

Cotangent is the reciprocal of tangent, representing the ratio of the x-coordinate to the y-coordinate. It is also negative in the second quadrant.

Summary of Trigonometric Values

In conclusion, given the point (-4, 7) on the terminal side of angle $\theta$ in standard position, the exact values of the six trigonometric functions are:

  • sinθ=76565{sin \theta = \frac{7\sqrt{65}}{65}}
  • cosθ=−46565{cos \theta = \frac{-4\sqrt{65}}{65}}
  • tanθ=−74{tan \theta = -\frac{7}{4}}
  • cscθ=657{csc \theta = \frac{\sqrt{65}}{7}}
  • secθ=−654{sec \theta = -\frac{\sqrt{65}}{4}}
  • cotθ=−47{cot \theta = -\frac{4}{7}}

These values provide a comprehensive understanding of the trigonometric relationships associated with the angle whose terminal side passes through the point (-4, 7).

Common Mistakes and How to Avoid Them

When calculating trigonometric function values, it's easy to make mistakes if you're not careful. Understanding common pitfalls can help you avoid them. Here are some of the most frequent errors and strategies to prevent them:

1. Incorrectly Applying the Pythagorean Theorem

The Pythagorean theorem is crucial for finding the distance r. A common mistake is to add or subtract the squares of x and y incorrectly. Always remember that r is the hypotenuse, so: r=x2+y2{r = \sqrt{x^2 + y^2}}

Ensure you're squaring both x and y correctly, and then taking the square root of their sum. Double-check your calculations to avoid simple arithmetic errors.

2. Forgetting the Sign of x and y

The signs of x and y are critical in determining the signs of the trigonometric functions. For example, in the second quadrant, x is negative and y is positive. If you forget to include these signs, you'll get incorrect trigonometric values. Always pay attention to the quadrant in which the terminal side lies and the corresponding signs of x and y.

3. Rationalizing Denominators Incorrectly

It's standard practice to rationalize denominators, but errors can occur during this process. Ensure you multiply both the numerator and the denominator by the same radical. For instance, to rationalize $\frac{7}{\sqrt{65}}$, multiply both the numerator and denominator by $\sqrt{65}$ to get $\frac{7\sqrt{65}}{65}$. Double-check that you've performed the multiplication correctly and simplified the fraction if necessary.

4. Using the Wrong Reciprocal Functions

Confusing the reciprocal trigonometric functions is another common mistake. Remember:

  • Cosecant (csc) is the reciprocal of sine (sin)
  • Secant (sec) is the reciprocal of cosine (cos)
  • Cotangent (cot) is the reciprocal of tangent (tan)

When calculating these values, double-check that you're using the correct reciprocal relationship.

5. Not Simplifying Fractions

Always simplify your final answers. If you end up with a fraction that can be reduced, make sure to do so. For example, if you calculate a trigonometric value as $\frac{2}{4}$, simplify it to $\frac{1}{2}$. Failure to simplify can lead to incorrect answers in subsequent calculations.

6. Misunderstanding the Definitions of Trigonometric Functions

Ensure you have a solid grasp of the definitions of the six trigonometric functions in terms of x, y, and r. Mix-ups like confusing sine and cosine or using the tangent formula incorrectly can lead to significant errors. Regularly review the definitions to reinforce your understanding.

7. Ignoring the Quadrant of the Angle

The quadrant in which the angle's terminal side lies affects the signs of the trigonometric functions. For instance, tangent is positive in the first and third quadrants but negative in the second and fourth. Always consider the quadrant to ensure your final answers have the correct signs.

Strategies to Avoid Mistakes

  • Write Down the Formulas: Before starting the calculations, write down the relevant formulas (Pythagorean theorem, trigonometric function definitions) to have them handy.
  • Double-Check Calculations: After each step, double-check your calculations to catch any arithmetic errors early on.
  • Draw a Diagram: Sketching the point (-4, 7) on the coordinate plane can help you visualize the problem and confirm the signs of the trigonometric functions.
  • Use a Reference Table: Keep a reference table of trigonometric function definitions and signs in different quadrants for quick reference.
  • Practice Regularly: Consistent practice is the best way to master trigonometric calculations and minimize mistakes.

By understanding these common mistakes and implementing the strategies to avoid them, you can increase your accuracy and confidence in calculating trigonometric function values.

Applications of Trigonometric Functions

The trigonometric functions are not just theoretical concepts; they have a wide range of practical applications in various fields. Understanding these applications can provide a deeper appreciation for the importance of trigonometry. This section will explore some key areas where trigonometric functions are used extensively.

1. Navigation and Surveying

Trigonometry is fundamental to navigation, both in maritime and aviation contexts. The sine, cosine, and tangent functions are used to calculate angles and distances, which are essential for determining a ship's or aircraft's position and course. Surveyors also use trigonometric principles to measure land areas, determine elevations, and create accurate maps. Techniques such as triangulation, which relies heavily on trigonometric ratios, allow surveyors to calculate distances and angles with precision.

2. Physics and Engineering

In physics, trigonometric functions are used to describe oscillatory motion, such as waves and vibrations. For instance, the motion of a pendulum or the propagation of light and sound waves can be modeled using sine and cosine functions. Engineers use trigonometry to analyze forces and stresses in structures, design bridges and buildings, and work with electrical circuits where alternating current (AC) waveforms are described using trigonometric functions. The analysis of vector components, crucial in mechanics and electromagnetism, also relies heavily on trigonometric principles.

3. Computer Graphics and Game Development

Trigonometry plays a vital role in computer graphics and game development. It is used to rotate, scale, and translate objects in 2D and 3D spaces. Trigonometric functions are essential for creating realistic animations, simulating camera movements, and calculating lighting and shading effects. In game development, trigonometric calculations are used for character movement, projectile trajectories, and collision detection, making the virtual world behave according to physical laws.

4. Astronomy

Astronomers use trigonometry to measure distances to stars and planets, track the movement of celestial bodies, and understand the structure of galaxies. Techniques such as parallax, which measures the apparent shift in a star's position due to Earth's orbit around the Sun, rely on trigonometric calculations. Trigonometry is also used in satellite navigation systems and in the study of celestial mechanics, which describes the motion of objects in space under the influence of gravity.

5. Music and Sound Analysis

Trigonometric functions are used in the analysis and synthesis of sound. Sound waves can be represented as trigonometric functions, and Fourier analysis, a technique that decomposes complex waveforms into their constituent frequencies, relies heavily on trigonometric principles. This is crucial in digital audio processing, music production, and the design of audio equipment. Musical instrument design also incorporates trigonometric concepts to optimize sound quality and resonance.

6. Financial Analysis

While not as direct as in physics or engineering, trigonometric functions can be used to model cyclical patterns in financial markets. Some economic models use trigonometric functions to represent seasonal variations in sales or fluctuations in interest rates. Time series analysis, a statistical method used to analyze data points collected over time, may also incorporate trigonometric components to capture periodic trends.

7. Medical Imaging

Trigonometry is used in various medical imaging techniques, such as X-ray computed tomography (CT) scans and magnetic resonance imaging (MRI). These techniques use mathematical algorithms to reconstruct 2D and 3D images of the body from multiple projections or signals. Trigonometric functions play a crucial role in these algorithms, allowing for the accurate representation of anatomical structures.

Real-World Examples

  • GPS Navigation: Global Positioning System (GPS) devices use trigonometry to calculate a user's position based on signals from multiple satellites. The distances to the satellites and the angles between them are used to pinpoint the user's location accurately.
  • Bridge Design: Engineers use trigonometric functions to calculate the forces and stresses acting on different parts of a bridge. This ensures the bridge's stability and safety under various load conditions.
  • Animation in Movies: Animated movies rely heavily on trigonometric functions to create realistic movements and visual effects. Characters and objects are rotated, translated, and scaled using trigonometric calculations.
  • Weather Forecasting: Meteorologists use trigonometric functions in weather models to predict temperature variations, wind patterns, and precipitation probabilities.

Conclusion

The ability to calculate the exact values of trigonometric functions given a point on the terminal side is a fundamental skill in trigonometry. By understanding the definitions of the trigonometric functions, the Pythagorean theorem, and the importance of considering the quadrant, you can confidently solve these types of problems. Furthermore, recognizing the diverse applications of trigonometric functions in various fields highlights the practical significance of this mathematical concept. Whether you're a student learning trigonometry or a professional applying it in your field, a solid understanding of these principles will serve you well.