Solve Sin Θ = Cos (θ + 20): Step-by-Step Guide

Hey guys! Let's dive into the fascinating world of trigonometry and tackle a classic problem: solving the equation sin θ = cos (θ + 20). Trigonometric equations might seem daunting at first, but with the right approach and a sprinkle of trigonometric identities, they become quite manageable. In this comprehensive guide, we'll break down the problem step-by-step, explore different methods to find solutions, and even peek at the general solutions that capture all possible angles. So, buckle up, grab your calculators, and let's get started!

Understanding the Basics

Before we jump into the nitty-gritty, let’s refresh our understanding of some fundamental trigonometric concepts. Trigonometry, at its heart, is the study of relationships between angles and sides of triangles. The three primary trigonometric functions—sine (sin), cosine (cos), and tangent (tan)—are the building blocks of this field. They describe the ratios of sides in a right-angled triangle. Specifically, sin θ is the ratio of the opposite side to the hypotenuse, cos θ is the ratio of the adjacent side to the hypotenuse, and tan θ is the ratio of the opposite side to the adjacent side.

The unit circle is another crucial concept. Imagine a circle with a radius of 1 unit centered at the origin of a coordinate plane. As a point moves around this circle, its x and y coordinates correspond to the cosine and sine of the angle formed with the positive x-axis, respectively. This visual representation helps us understand how trigonometric functions vary as angles change. For instance, sine and cosine functions oscillate between -1 and 1, reflecting the bounded nature of the unit circle’s coordinates. Additionally, understanding the periodicity of trigonometric functions is vital. Both sine and cosine functions repeat their values every 360 degrees (or 2π radians). This periodicity leads to multiple solutions for trigonometric equations, as we'll see later.

Transforming the Equation

Now, let's focus on our main problem: sin θ = cos (θ + 20). The key to solving this equation lies in recognizing and applying trigonometric identities. These identities are equations that are always true for any value of the angles involved. One identity that's particularly useful here is the cofunction identity. Cofunction identities show the relationship between trigonometric functions of complementary angles. Complementary angles are angles that add up to 90 degrees (or π/2 radians). The cofunction identity we’ll use states that sin θ = cos (90° - θ).

Using this identity, we can rewrite the left side of our equation. Instead of sin θ, we can write cos (90° - θ). Our equation now looks like this: cos (90° - θ) = cos (θ + 20). This transformation is crucial because it allows us to compare cosine functions directly. When two cosine functions are equal, it means that the angles inside the cosine functions are either equal or differ by a multiple of 360 degrees. In other words, if cos A = cos B, then A = B + 360°k or A = -B + 360°k, where k is an integer. This property stems from the periodic nature of the cosine function. Understanding this principle will help us find all possible solutions to our equation.

Solving the Equation

With our equation transformed to cos (90° - θ) = cos (θ + 20), we can now proceed to solve for θ. As we discussed, if cos A = cos B, then A = B + 360°k or A = -B + 360°k, where k is an integer. Applying this to our equation, we get two cases:

Case 1: 90° - θ = θ + 20° + 360°k

Let's simplify this equation. First, we'll move all the θ terms to one side and the constants to the other side. Adding θ to both sides gives us 90° = 2θ + 20° + 360°k. Next, we'll subtract 20° from both sides, resulting in 70° = 2θ + 360°k. Now, we divide the entire equation by 2 to isolate θ. This gives us θ = 35° - 180°k. This is a general solution that represents an infinite set of angles, each corresponding to a different integer value of k. For example, if k = 0, we get θ = 35°. If k = 1, we get θ = 35° - 180° = -145°. Each value of k provides a valid solution to the original equation.

Case 2: 90° - θ = -(θ + 20°) + 360°k

This case involves the negative of the second angle. Distributing the negative sign, we get 90° - θ = -θ - 20° + 360°k. Now, let’s simplify. Adding θ to both sides gives us 90° = -20° + 360°k. Next, we add 20° to both sides, resulting in 110° = 360°k. To solve for k, we divide both sides by 360°, which gives us k = 110°/360° = 11/36. However, since k must be an integer, this case does not yield any valid solutions. This is an important observation because it highlights that not all potential solution paths lead to actual solutions. It's crucial to check each case carefully.

General Solutions and Specific Solutions

From Case 1, we found the general solution θ = 35° - 180°k. This general solution provides a framework for finding infinitely many specific solutions. The integer k acts as a parameter that generates different angles that satisfy the equation. For instance, when k = 0, we get θ = 35°. When k = 1, we get θ = 35° - 180° = -145°. When k = -1, we get θ = 35° + 180° = 215°. These are just a few examples; we can generate many more by plugging in different integer values for k.

Sometimes, we're interested in solutions within a specific interval, such as 0° ≤ θ < 360°. To find these specific solutions, we need to identify the values of k that produce angles within this interval. Let's consider our general solution θ = 35° - 180°k. When k = 0, we have θ = 35°, which falls within our interval. When k = -1, we have θ = 35° + 180° = 215°, which also falls within our interval. However, when k = 1, we have θ = 35° - 180° = -145°, which is outside our interval. Similarly, when k = -2, we have θ = 35° + 360° = 395°, which is also outside our interval. Therefore, the specific solutions within the interval 0° ≤ θ < 360° are 35° and 215°.

Graphical Verification

Another powerful way to verify our solutions is by using a graphical approach. We can plot the graphs of y = sin θ and y = cos (θ + 20) on the same coordinate plane. The points where these graphs intersect represent the solutions to our equation sin θ = cos (θ + 20). By visually inspecting the graphs, we can confirm the solutions we found algebraically.

Using graphing software or a graphing calculator, plot these two functions. You'll notice that the graphs intersect at multiple points. By zooming in and tracing the intersection points, you can identify the angles where the two functions have the same value. Within the interval 0° ≤ θ < 360°, you’ll see the intersections occur at approximately θ = 35° and θ = 215°, which aligns perfectly with our algebraic solutions. This graphical verification provides a strong confirmation of our results and adds an intuitive layer to our understanding of the problem.

Alternative Methods

While using the cofunction identity is a straightforward method, there are alternative approaches to solving this equation. One such method involves using the sine addition formula. The sine addition formula states that sin (A + B) = sin A cos B + cos A sin B. We can rewrite our original equation sin θ = cos (θ + 20) by applying the cofunction identity to cos (θ + 20) first. We know that cos (x) = sin (90° - x), so we can write cos (θ + 20) as sin (90° - (θ + 20)) = sin (70° - θ). Our equation now becomes sin θ = sin (70° - θ).

When two sine functions are equal, it means that the angles inside the sine functions are either equal or supplementary (add up to 180 degrees), modulo 360 degrees. In other words, if sin A = sin B, then A = B + 360°k or A = 180° - B + 360°k, where k is an integer. Applying this to our equation, we get two cases:

Case 1: θ = 70° - θ + 360°k

Adding θ to both sides gives us 2θ = 70° + 360°k. Dividing by 2, we get θ = 35° + 180°k. This solution is similar to what we found earlier, but with a slight difference in the sign of the 180°k term. This difference arises because we're using a different approach, but the solutions are fundamentally the same.

Case 2: θ = 180° - (70° - θ) + 360°k

Simplifying, we get θ = 180° - 70° + θ + 360°k, which simplifies further to θ = 110° + θ + 360°k. Subtracting θ from both sides gives us 0 = 110° + 360°k. This equation implies that k = -110°/360°, which is not an integer, so this case does not yield any valid solutions.

By using the sine addition formula, we arrive at the same set of solutions as we did with the cofunction identity, demonstrating that there are often multiple paths to the same destination in trigonometry.

Conclusion

Solving trigonometric equations like sin θ = cos (θ + 20) involves a combination of trigonometric identities, algebraic manipulation, and a solid understanding of the periodic nature of trigonometric functions. We started by transforming the equation using the cofunction identity, which allowed us to compare cosine functions directly. We then derived general solutions and identified specific solutions within a given interval. We also verified our results graphically, providing a visual confirmation of our algebraic solutions. Furthermore, we explored an alternative method using the sine addition formula, highlighting the versatility of trigonometric problem-solving.

Trigonometry is a beautiful and powerful tool in mathematics, with applications spanning from navigation and engineering to physics and computer graphics. By mastering trigonometric identities and problem-solving techniques, you’ll unlock a deeper appreciation for this field and its real-world applications. So, keep practicing, keep exploring, and never stop asking questions. You've got this!