Hey everyone! Today, we're diving deep into a fun trigonometric problem. Imagine you're given that 5 cos θ = -3 and θ falls within the range of 0° to 180°. Our mission? To find the values of 1 - cot² θ and sec θ + sin² θ without reaching for that calculator. Sounds like a challenge, right? But trust me, it's going to be an exciting journey through the world of trigonometry!
Understanding the Basics
Before we jump into solving, let's quickly recap some essential trigonometric concepts. Remember, cosine (cos θ), cotangent (cot θ), secant (sec θ), and sine (sin θ) are all linked together through the sides of a right-angled triangle. Cosine is the ratio of the adjacent side to the hypotenuse, sine is the ratio of the opposite side to the hypotenuse, cotangent is the ratio of the adjacent side to the opposite side, and secant is the reciprocal of cosine, meaning it's the ratio of the hypotenuse to the adjacent side. Got it? Great!
Now, let's break down what we're given. We know that 5 cos θ = -3. To make things simpler, let's isolate cos θ. Divide both sides by 5, and we get cos θ = -3/5. This is a crucial piece of information. Since cos θ is negative and θ lies between 0° and 180°, we know that θ must be in the second quadrant. Why? Because cosine is negative in the second and third quadrants, but our angle is limited to the first two quadrants. This is where the diagram comes into play – it helps us visualize the problem.
Drawing the Diagram
Let’s sketch a right-angled triangle in the second quadrant. Imagine the x-axis and y-axis forming the familiar Cartesian plane. Draw a line from the origin into the second quadrant. This line is our hypotenuse. Now, drop a perpendicular line from the end of the hypotenuse to the x-axis. Voila! We have a right-angled triangle. The angle θ is formed between the negative x-axis and the hypotenuse. Since cos θ = -3/5, we can label the adjacent side as -3 and the hypotenuse as 5. Note that the hypotenuse is always positive because it represents a distance.
To find the opposite side, we'll use the Pythagorean theorem: a² + b² = c², where a and b are the sides of the triangle, and c is the hypotenuse. In our case, (-3)² + b² = 5². This simplifies to 9 + b² = 25. Subtract 9 from both sides, and we get b² = 16. Taking the square root, we find that b = 4. So, the opposite side is 4. Now our triangle is complete! We have all three sides: adjacent = -3, opposite = 4, and hypotenuse = 5. Remember, the negative sign for the adjacent side is crucial because it indicates the direction in the Cartesian plane.
Solving for 1 - cot² θ
Okay, with our triangle in place, let's tackle the first part of the problem: finding the value of 1 - cot² θ. Remember that cot θ is the ratio of the adjacent side to the opposite side. In our triangle, this means cot θ = -3/4. Now, we need to find cot² θ. This simply means squaring cot θ, so cot² θ = (-3/4)² = 9/16. Now we can substitute this value into our expression: 1 - cot² θ = 1 - 9/16. To subtract these, we need a common denominator. We can rewrite 1 as 16/16. So, 1 - 9/16 = 16/16 - 9/16 = 7/16. And there you have it! The value of 1 - cot² θ is 7/16. Isn't it satisfying when things come together like that?
Unraveling sec θ + sin² θ
Now, let's move on to the second part: finding the value of sec θ + sin² θ. We already know that cos θ = -3/5. Since sec θ is the reciprocal of cos θ, we can simply flip the fraction. So, sec θ = -5/3. Next, we need to find sin θ. Remember, sin θ is the ratio of the opposite side to the hypotenuse. In our triangle, this means sin θ = 4/5. Now, we need to find sin² θ. Squaring sin θ, we get sin² θ = (4/5)² = 16/25. Now we can substitute these values into our expression: sec θ + sin² θ = -5/3 + 16/25. To add these fractions, we need a common denominator. The least common multiple of 3 and 25 is 75. So, we rewrite the fractions with the denominator 75: -5/3 = -125/75 and 16/25 = 48/75. Now we can add them: -125/75 + 48/75 = -77/75. And that's it! The value of sec θ + sin² θ is -77/75.
Wrapping Up Trigonometric Solutions
Wow, we've really tackled a challenging trigonometric problem head-on! We started with 5 cos θ = -3 and θ in the range of 0° to 180°, and without using a calculator, we successfully found the values of 1 - cot² θ and sec θ + sin² θ. We used our understanding of trigonometric ratios, the Pythagorean theorem, and the importance of visualizing the problem with a diagram. Remember, guys, trigonometry might seem daunting at first, but with practice and a solid understanding of the basics, you can conquer any trigonometric mystery. Keep exploring, keep practicing, and most importantly, have fun with math!
Essential Trigonometry Problem If 5 cos θ=-3 and θ ∈ [0°; 180°]
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Introduction to the Problem
Trigonometry, a cornerstone of mathematics, often presents intriguing challenges that require a blend of conceptual understanding and problem-solving skills. Solving trigonometric problems can sometimes feel like navigating a maze, but with the right tools and techniques, even the most complex equations can be解碼ed. One such problem involves finding the values of trigonometric expressions given specific conditions. In this article, we will dissect a classic trigonometry question: If 5 cos θ = -3 and θ is within the interval [0°; 180°], determine the values of 1 - cot² θ and sec θ + sin² θ without the aid of a calculator. This exercise is an excellent opportunity to reinforce our understanding of trigonometric ratios, the unit circle, and the Pythagorean theorem.
Understanding the Given Condition: 5 cos θ = -3
The first step in tackling any trigonometric problem is to understand the given conditions. We are presented with the equation 5 cos θ = -3. This tells us something crucial about the angle θ. Let's isolate cos θ by dividing both sides by 5, giving us cos θ = -3/5. The fact that cos θ is negative is our first clue. In the realm of trigonometry, the sign of the cosine function tells us which quadrants the angle θ might lie in.
To fully appreciate this, we need to recall the behavior of trigonometric functions in different quadrants. The unit circle is an invaluable tool for this purpose. Imagine a circle with a radius of 1 unit centered at the origin of a coordinate plane. The cosine of an angle corresponds to the x-coordinate of the point where the terminal side of the angle intersects the circle, while the sine corresponds to the y-coordinate. In the first quadrant (0° to 90°), both cosine and sine are positive. In the second quadrant (90° to 180°), sine is positive, but cosine is negative. In the third quadrant (180° to 270°), both sine and cosine are negative. And finally, in the fourth quadrant (270° to 360°), cosine is positive, but sine is negative. Given that cos θ = -3/5, a negative value, and the condition θ ∈ [0°; 180°], we can confidently deduce that θ must lie in the second quadrant. This crucial piece of information guides the rest of our solution.
Visualizing with a Right-Angled Triangle
To further解碼 the problem, let's visualize it using a right-angled triangle. This is a common technique in solving trigonometric problems. Since θ is in the second quadrant, we'll construct a reference triangle in this quadrant. Draw a triangle such that the angle θ is formed between the negative x-axis and the hypotenuse. The adjacent side will lie along the negative x-axis, the opposite side will be along the positive y-axis, and the hypotenuse will connect the two. Knowing that cos θ = -3/5, we can label the adjacent side as -3 and the hypotenuse as 5. Note that the hypotenuse is always positive, representing a distance.
Now, we need to find the length of the opposite side. This is where the Pythagorean theorem comes into play. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²). In our case, we have (-3)² + b² = 5². This simplifies to 9 + b² = 25. Subtracting 9 from both sides, we get b² = 16. Taking the square root, we find b = 4. So, the opposite side has a length of 4. With all three sides of the triangle now known (adjacent = -3, opposite = 4, hypotenuse = 5), we are well-equipped to find the values of other trigonometric functions.
Calculating 1 - cot² θ
The first expression we need to evaluate is 1 - cot² θ. To do this, we first need to determine the value of cot θ. Recall that cot θ is the reciprocal of tan θ, and tan θ is the ratio of the opposite side to the adjacent side. Therefore, cot θ is the ratio of the adjacent side to the opposite side. In our triangle, this means cot θ = -3/4. Now, we need to find cot² θ. Squaring cot θ, we get cot² θ = (-3/4)² = 9/16. Substituting this value into our expression, we have 1 - cot² θ = 1 - 9/16. To subtract these, we need a common denominator. We can rewrite 1 as 16/16. So, 1 - 9/16 = 16/16 - 9/16 = 7/16. Thus, the value of 1 - cot² θ is 7/16.
Determining sec θ + sin² θ
Next, we need to find the value of sec θ + sin² θ. We already know that cos θ = -3/5. Since sec θ is the reciprocal of cos θ, we simply flip the fraction. So, sec θ = -5/3. Now, let's find sin θ. Remember, sin θ is the ratio of the opposite side to the hypotenuse. In our triangle, this means sin θ = 4/5. We now need to calculate sin² θ. Squaring sin θ, we get sin² θ = (4/5)² = 16/25. Substituting these values into our expression, we have sec θ + sin² θ = -5/3 + 16/25. To add these fractions, we need a common denominator. The least common multiple of 3 and 25 is 75. We rewrite the fractions with the denominator 75: -5/3 = -125/75 and 16/25 = 48/75. Adding them gives us -125/75 + 48/75 = -77/75. Therefore, the value of sec θ + sin² θ is -77/75.
Conclusion Trigonometry Mastery
In conclusion, we have successfully navigated through a challenging trigonometric problem without the use of a calculator. We started with the given condition 5 cos θ = -3 and θ ∈ [0°; 180°], and we skillfully determined the values of 1 - cot² θ and sec θ + sin² θ. This journey involved understanding the behavior of trigonometric functions in different quadrants, visualizing the problem with a right-angled triangle, applying the Pythagorean theorem, and utilizing the definitions of cot θ, sec θ, and sin θ.
Mastering trigonometry requires consistent practice and a solid grasp of fundamental concepts. By breaking down problems into manageable steps and leveraging tools like the unit circle and trigonometric ratios, you can confidently tackle even the most intricate equations. Remember, the key to solving trigonometric problems lies in understanding the relationships between the functions and applying the appropriate techniques. Keep practicing, and you'll become a trigonometry pro in no time!
Mastering Trigonometry Solving for 1 - cot² θ and sec θ + sin² θ
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Introduction Unveiling Trigonometric Equations
Trigonometry, often perceived as a complex branch of mathematics, is fundamentally about the relationships between angles and sides in triangles. Solving trigonometric equations is a crucial skill in various fields, from physics and engineering to navigation and computer graphics. These problems require a blend of conceptual understanding, algebraic manipulation, and a strong grasp of trigonometric identities. In this article, we'll tackle a specific problem that exemplifies these skills: Given that 5 cos θ = -3 and θ lies within the interval [0°, 180°], we aim to determine the values of 1 - cot² θ and sec θ + sin² θ without relying on a calculator. This exercise will not only solidify our understanding of basic trigonometric functions but also highlight the power of visual aids and logical deduction in mathematical problem-solving.
Deciphering the Initial Condition 5 cos θ = -3
The cornerstone of any mathematical solution lies in understanding the initial conditions. We're given that 5 cos θ = -3. To simplify this, let's isolate cos θ by dividing both sides of the equation by 5. This yields cos θ = -3/5. This seemingly simple result carries significant implications about the nature of the angle θ. The fact that cos θ is negative immediately narrows down the possible quadrants in which θ can reside.
Recall the behavior of cosine in the four quadrants of the coordinate plane. The unit circle provides an excellent visual aid for this. In the first quadrant (0° to 90°), where both x and y coordinates are positive, cosine is positive. In the second quadrant (90° to 180°), where x is negative and y is positive, cosine is negative. The third quadrant (180° to 270°) sees both x and y as negative, making cosine negative as well. Finally, in the fourth quadrant (270° to 360°), with positive x and negative y, cosine is positive. Since our cos θ is negative and we're given the constraint θ ∈ [0°, 180°], we can definitively conclude that θ must lie in the second quadrant. This eliminates the possibility of θ being in the first quadrant, simplifying our task considerably. The second crucial piece of information is the value itself, -3/5, which suggests the ratio of the adjacent side to the hypotenuse in a related right triangle.
Constructing the Right Triangle A Visual Approach
A visual representation can often illuminate the path to a solution. To visualize our problem, we'll construct a right triangle in the second quadrant. Draw a line segment from the origin into the second quadrant, representing the hypotenuse. From the end of this segment, drop a perpendicular to the x-axis, creating a right triangle. The angle θ is formed between the negative x-axis and the hypotenuse. Since cos θ = -3/5, we can label the adjacent side as -3 and the hypotenuse as 5. It's essential to remember that the hypotenuse is always a positive value, representing the distance from the origin. The negative sign on the adjacent side is crucial, as it signifies the direction along the x-axis in the coordinate plane.
Now, we need to determine the length of the opposite side. Here, the Pythagorean theorem comes to our rescue. This fundamental theorem states that in a right triangle, the square of the hypotenuse (c²) is equal to the sum of the squares of the other two sides (a² + b²). In our case, we have (-3)² + b² = 5². Simplifying, we get 9 + b² = 25. Subtracting 9 from both sides gives us b² = 16. Taking the square root, we find b = 4. Thus, the opposite side has a length of 4. With all three sides now known (adjacent = -3, opposite = 4, hypotenuse = 5), we have a complete reference triangle, allowing us to calculate other trigonometric ratios.
Calculating 1 - cot² θ Navigating Trigonometric Identities
The first expression we need to evaluate is 1 - cot² θ. To find this, we first need to determine the value of cot θ. Recall that cotangent (cot θ) is the reciprocal of tangent (tan θ). In a right triangle, tan θ is defined as the ratio of the opposite side to the adjacent side. Therefore, cot θ is the ratio of the adjacent side to the opposite side. In our triangle, this means cot θ = -3/4. Now, we need to find cot² θ. This simply involves squaring the value of cot θ: cot² θ = (-3/4)² = 9/16. Substituting this into our expression, we have 1 - cot² θ = 1 - 9/16. To perform the subtraction, we need a common denominator. We can rewrite 1 as 16/16. So, 1 - 9/16 = 16/16 - 9/16 = 7/16. Therefore, the value of 1 - cot² θ is 7/16.
Determining sec θ + sin² θ Combining Trigonometric Functions
Next, we need to evaluate the expression sec θ + sin² θ. We already know that cos θ = -3/5. Since secant (sec θ) is the reciprocal of cosine (cos θ), we can find sec θ by simply flipping the fraction: sec θ = -5/3. Now, let's find sin θ. Sine (sin θ) is defined as the ratio of the opposite side to the hypotenuse. In our triangle, this means sin θ = 4/5. To find sin² θ, we square the value of sin θ: sin² θ = (4/5)² = 16/25. Substituting these values into our expression, we have sec θ + sin² θ = -5/3 + 16/25. To add these fractions, we need a common denominator. The least common multiple of 3 and 25 is 75. Rewriting the fractions with the common denominator, we get -5/3 = -125/75 and 16/25 = 48/75. Now we can add: -125/75 + 48/75 = -77/75. Therefore, the value of sec θ + sin² θ is -77/75.
Conclusion Mastering Trigonometric Problem-Solving
We have successfully navigated a complex trigonometry problem without the aid of a calculator, demonstrating a strong understanding of trigonometric functions, ratios, and the Pythagorean theorem. Starting with the condition 5 cos θ = -3 and θ ∈ [0°, 180°], we methodically determined the values of 1 - cot² θ and sec θ + sin² θ. This process involved identifying the correct quadrant, constructing a reference right triangle, and applying the definitions of cotangent, secant, and sine. By breaking down the problem into manageable steps and utilizing visual aids, we were able to arrive at the solutions confidently.
Solving trigonometric equations requires a blend of conceptual knowledge and problem-solving skills. By consistently practicing and reinforcing these fundamental principles, you can master trigonometry and confidently tackle a wide range of mathematical challenges. Remember, the key to success lies in a deep understanding of the relationships between trigonometric functions and their geometric interpretations. Keep exploring, keep practicing, and watch your trigonometry skills soar!