Trigonometry Problem Solutions: Find Sine, Cosine, And Values

Right Triangle Trigonometry: Finding sin(90° - θ) and cos(90° - θ) when sin θ = 5/13

Hey guys! Let's dive into a classic trigonometry problem. We've got a right triangle ABC, and we know that sin θ = 5/13. Our mission, should we choose to accept it (and we do!), is to find the values of sin(90° - θ) and cos(90° - θ). This problem really hammers home the relationships between sine, cosine, and complementary angles, which are super important in trigonometry. So, let’s get started and break this down step by step.

First off, remember what sine means in a right triangle. Sine (sin θ) is defined as the ratio of the length of the side opposite the angle θ to the length of the hypotenuse. In our case, sin θ = 5/13 tells us that if the side opposite θ is 5 units long, then the hypotenuse is 13 units long. Think of the classic 5-12-13 right triangle; it’s a Pythagorean triple that fits this scenario perfectly. We're going to use this information and the Pythagorean theorem to figure out the adjacent side.

Now, let's use the Pythagorean theorem to find the length of the side adjacent to θ. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b), or a² + b² = c². In our triangle, let’s call the side opposite θ 'a', the side adjacent to θ 'b', and the hypotenuse 'c'. We know a = 5 and c = 13, so we can plug these values into the theorem: 5² + b² = 13². This simplifies to 25 + b² = 169. Subtracting 25 from both sides gives us b² = 144. Taking the square root of both sides, we find that b = 12. Awesome! We now know the length of the side adjacent to θ is 12 units.

Next up, let's tackle sin(90° - θ). Here’s where the magic of complementary angles comes in. Remember that in a right triangle, the two acute angles (θ and 90° - θ) are complementary, meaning they add up to 90°. A key trigonometric identity tells us that the sine of an angle is equal to the cosine of its complement. In other words, sin(90° - θ) = cos θ. So, to find sin(90° - θ), we just need to find cos θ. Cosine (cos θ) is defined as the ratio of the length of the side adjacent to θ to the length of the hypotenuse. We already found that the side adjacent to θ is 12 units long, and we know the hypotenuse is 13 units long. Therefore, cos θ = 12/13. And guess what? That means sin(90° - θ) = 12/13!

Finally, let's find cos(90° - θ). Using a similar logic as above, we can use another trigonometric identity: the cosine of an angle is equal to the sine of its complement. So, cos(90° - θ) = sin θ. Lucky for us, we already know that sin θ = 5/13. Therefore, cos(90° - θ) = 5/13. We did it! We successfully found both sin(90° - θ) and cos(90° - θ) using our knowledge of right triangles, the Pythagorean theorem, and complementary angle identities.

In summary, by understanding the fundamental trigonometric ratios and the relationships between complementary angles, we can solve some pretty cool problems. Remember to always visualize the triangle, use the Pythagorean theorem to find missing sides, and keep those trigonometric identities handy. You'll be a trigonometry whiz in no time!

Determining the Value of sin 25°

Okay, guys, let's tackle another trig problem: determining the value of sin 25°. Now, this one's a bit different because 25° isn't one of our special angles (like 30°, 45°, or 60°) where we can easily recall exact values. So, we’re not going to get a nice, neat fraction or radical expression. Instead, we'll need to rely on a calculator or trigonometric tables to find an approximate decimal value. This is a common situation in real-world applications where angles aren't always those perfect, textbook values. So, let’s jump in and see how we can approach this.

First off, it’s important to recognize that sin 25° represents the sine of an angle that isn't a standard angle for which we typically memorize values. We know the sine values for 0°, 30°, 45°, 60°, and 90°, but 25° falls in between. This means we won't be able to express sin 25° using simple radicals or fractions like we do for those special angles. Instead, we need to find a way to approximate its value. That's where our trusty calculators or trigonometric tables come into play.

Now, let's talk about how we'd use a calculator. Most scientific calculators have a sin function key. To find sin 25°, you simply need to enter '25' and then press the 'sin' button. Make sure your calculator is set to degree mode (DEG), not radians (RAD), because we're working with degrees here. If your calculator is in radian mode, you'll get a completely different answer! Once you hit the 'sin' button, the calculator will display an approximate decimal value for sin 25°. It’s super straightforward, but double-checking the mode setting is crucial to avoid errors.

If you don't have a calculator handy, you can use trigonometric tables. These tables list the sine, cosine, and tangent values for various angles, usually in increments of degrees or fractions of degrees. To find sin 25° in a trig table, you’d locate the row corresponding to 25° and then find the value in the sine column. Trigonometric tables are a bit old-school, but they're still a useful tool to know about, especially if you're ever in a situation where you don't have access to a calculator. Plus, they give you a good visual understanding of how the sine function changes as the angle increases.

When you use a calculator to find sin 25°, you’ll get an approximate value of 0.4226 (rounded to four decimal places). This value tells us the ratio of the opposite side to the hypotenuse in a right triangle with a 25° angle. It’s a decimal because, as we discussed, 25° isn't a special angle with a simple fractional or radical sine value. This is totally normal, and in many practical situations, a decimal approximation is exactly what we need.

So, in summary, while we can't find an exact, simplified value for sin 25° using our knowledge of special angles, we can easily find an approximate value using a calculator or trigonometric tables. The key is to understand that some angles will require these tools, and that's perfectly okay. Just make sure your calculator is in the correct mode, and you'll be golden! This highlights the importance of knowing when to use different tools in trigonometry, and how to interpret the results we get.

Finding the Value of f

Alright, let's switch gears a bit, guys. Our next task is to find the value of f. Now, without any additional context, like an equation, a diagram, or a specific problem setup, we're flying a little blind here. The value of 'f' could be anything! It might represent an angle, a side length, a function output, or something else entirely. So, we need to make some educated guesses about what kind of problem we might be dealing with and how we can approach finding the value of 'f'. Let's explore some common scenarios and how we'd tackle them.

First, let's imagine 'f' is part of a trigonometric equation. Maybe we have something like sin(f) = 0.5 or cos(f) = √3/2. In this case, we'd be looking for an angle 'f' that satisfies the equation. To solve this, we'd need to recall our unit circle and special angles. For example, if sin(f) = 0.5, we know that f could be 30° (or π/6 radians) or 150° (or 5π/6 radians), since sine corresponds to the y-coordinate on the unit circle. We might also need to consider general solutions, adding multiples of 360° (or 2π radians) to our initial solutions to account for all possible angles. Trigonometric equations can be a lot of fun, but they require a solid understanding of trig functions and the unit circle.

Another possibility is that 'f' represents a side length in a triangle. If we're given a triangle with some known sides and angles, we might need to use the Law of Sines or the Law of Cosines to find 'f'. The Law of Sines is used when we have information about opposite side-angle pairs, while the Law of Cosines is useful when we have three sides or two sides and the included angle. For instance, if we have a triangle ABC with sides a, b, c and angles A, B, C, and we're trying to find side 'f' (let's say it's side 'a'), we might use the Law of Cosines: a² = b² + c² - 2bc * cos(A). We'd plug in the known values and solve for 'a' (which is 'f' in this case). Triangle problems are a staple in trigonometry, and mastering these laws is essential.

'f' could also represent the output of a function. For example, we might have a function like f(x) = 2x² - 3x + 1, and we're asked to find f(2). To do this, we simply substitute 2 for x in the function: f(2) = 2(2)² - 3(2) + 1 = 8 - 6 + 1 = 3. So, in this case, the value of f (when x = 2) is 3. Function evaluation is a fundamental skill in algebra and calculus, and it's important to be comfortable with substituting values into different types of functions.

Without more information, it's tough to pinpoint the exact method we should use to find the value of 'f'. But by considering these common scenarios – trigonometric equations, triangle side lengths, and function outputs – we can start to narrow down the possibilities. The key is to carefully examine the given problem and identify the relevant information and relationships. Once we have a clearer picture of the context, we can apply the appropriate techniques to solve for 'f'. Remember, guys, math is all about problem-solving, so don't be afraid to explore different approaches!

Finding the Value of m

Okay, let's move on to our next challenge: finding the value of m. Just like with 'f', we're operating with limited information here. The variable 'm' could represent a variety of things depending on the context of the problem. It could be a slope, a mass, a midpoint coordinate, or something else entirely. To figure out how to find 'm', we need to consider some possible scenarios and the mathematical tools we'd use in each case. Let's put on our thinking caps and explore some common situations where we might encounter 'm'.

One common scenario is when 'm' represents the slope of a line. In coordinate geometry, the slope of a line describes its steepness and direction. If we're given two points on the line, say (x₁, y₁) and (x₂, y₂), we can calculate the slope 'm' using the formula: m = (y₂ - y₁) / (x₂ - x₁). This formula tells us the change in the y-coordinate divided by the change in the x-coordinate, which gives us the