Finding Cos(α+β) With Sin(α) And Cos(β) In Quadrants
Alright, math enthusiasts! Let's dive into a trigonometric problem where we need to find the exact value of cos(α+β). We're given that angle α's terminal side lies in Quadrant I, and angle β's terminal side lies in Quadrant IV. We also know that sin(α) = 4/5 and cos(β) = 6/√61. Our mission, should we choose to accept it (and we do!), is to find cos(α+β).
Breaking Down the Problem
Before we jump into calculations, let's take a moment to understand what we're dealing with. This problem combines several key concepts in trigonometry:
- Quadrantal Angles: Knowing which quadrant an angle lies in helps us determine the signs of trigonometric functions. Remember, in Quadrant I, all trigonometric functions are positive. In Quadrant IV, cosine and secant are positive, while sine, cosecant, tangent, and cotangent are negative.
- Trigonometric Identities: We'll need to use the cosine addition formula: cos(α+β) = cos(α)cos(β) - sin(α)sin(β).
- Pythagorean Identity: This identity (sin²(θ) + cos²(θ) = 1) is our best friend for finding missing trigonometric values.
Now that we have a game plan, let's get to work!
Step-by-Step Solution
1. Finding cos(α)
We know sin(α) = 4/5, and we need to find cos(α). Since α is in Quadrant I, cos(α) will be positive. Let's use the Pythagorean identity:
sin²(α) + cos²(α) = 1
(4/5)² + cos²(α) = 1
16/25 + cos²(α) = 1
cos²(α) = 1 - 16/25
cos²(α) = 9/25
cos(α) = √(9/25)
cos(α) = 3/5
2. Finding sin(β)
We're given cos(β) = 6/√61, and we need to find sin(β). Since β is in Quadrant IV, sin(β) will be negative. Again, we'll use the Pythagorean identity:
sin²(β) + cos²(β) = 1
sin²(β) + (6/√61)² = 1
sin²(β) + 36/61 = 1
sin²(β) = 1 - 36/61
sin²(β) = 25/61
sin(β) = -√(25/61) (Remember, sine is negative in Quadrant IV)
sin(β) = -5/√61
3. Applying the Cosine Addition Formula
Now that we have all the pieces, let's plug them into the cosine addition formula:
cos(α+β) = cos(α)cos(β) - sin(α)sin(β)
cos(α+β) = (3/5)(6/√61) - (4/5)(-5/√61)
cos(α+β) = 18/(5√61) + 20/(5√61)
cos(α+β) = 38/(5√61)
4. Rationalizing the Denominator (Optional but Recommended)
To get rid of the square root in the denominator, we'll multiply both the numerator and denominator by √61:
cos(α+β) = (38/(5√61)) * (√61/√61)
cos(α+β) = (38√61) / (5 * 61)
cos(α+β) = (38√61) / 305
Final Answer
Therefore, the exact value of cos(α+β) is (38√61) / 305. Great job, guys! We tackled this trig problem like pros, using our knowledge of quadrantal angles, trigonometric identities, and the Pythagorean theorem.
Visualizing the Problem with Trigonometric Circles
To really nail down these concepts, let's visualize what's happening on the unit circle. The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It’s a fantastic tool for understanding trigonometric functions because it directly relates angles to coordinates.
Quadrant I and Angle α
Imagine angle α in Quadrant I. Since sin(α) = 4/5, this represents the y-coordinate of the point where the terminal side of α intersects the unit circle. We found that cos(α) = 3/5, which is the x-coordinate of that same point. So, we have a point (3/5, 4/5) on the unit circle corresponding to angle α.
Quadrant IV and Angle β
Now picture angle β in Quadrant IV. We know cos(β) = 6/√61, which is the x-coordinate. We calculated sin(β) = -5/√61, the y-coordinate. So, for angle β, we have the point (6/√61, -5/√61) on the unit circle.
The Cosine Addition Formula in Action
The cosine addition formula, cos(α+β) = cos(α)cos(β) - sin(α)sin(β), essentially helps us find the cosine of the angle formed by adding α and β. Geometrically, this is like rotating from the positive x-axis by angle α, and then further rotating by angle β.
By plugging in the values we found, we're using the coordinates of these points on the unit circle to calculate the x-coordinate of the point corresponding to the angle (α+β). This visualization can make the formula feel less abstract and more connected to geometric concepts.
Common Mistakes and How to Avoid Them
Trigonometry can be tricky, and it’s easy to make mistakes. Let’s go over some common pitfalls and how to avoid them:
Forgetting the Quadrant
The most common mistake is forgetting to consider the quadrant when determining the sign of trigonometric functions. Remember:
- Quadrant I: All positive
- Quadrant II: Sine (and cosecant) positive
- Quadrant III: Tangent (and cotangent) positive
- Quadrant IV: Cosine (and secant) positive
Always double-check which quadrant your angle is in before deciding whether sine, cosine, or tangent should be positive or negative.
Mixing Up Identities
There are a lot of trigonometric identities, and it’s easy to mix them up. Make sure you know the fundamental identities like the Pythagorean identity (sin²(θ) + cos²(θ) = 1) and the addition formulas (cos(α+β), sin(α+β), etc.). Creating a cheat sheet or flashcards can be super helpful.
Algebraic Errors
Simple algebraic errors, like messing up signs or making mistakes while squaring or taking square roots, can throw off your entire solution. Take your time, write out each step clearly, and double-check your work.
Not Rationalizing the Denominator
While not always required, rationalizing the denominator (getting rid of square roots in the denominator) is generally good practice. It makes your answer look cleaner and can sometimes make further calculations easier. If you're asked for the exact value, rationalizing is often expected.
Tips for Mastering Trigonometry
- Practice, Practice, Practice: The more problems you solve, the more comfortable you’ll become with the concepts and techniques.
- Visualize: Use the unit circle to visualize angles and trigonometric functions. This can make the relationships clearer and help you remember them.
- Know Your Identities: Memorize the key trigonometric identities and practice using them in different contexts.
- Check Your Work: Always double-check your calculations and make sure your answer makes sense in the context of the problem.
- Ask for Help: If you're stuck, don't hesitate to ask your teacher, classmates, or online resources for help. Everyone struggles sometimes, and getting another perspective can be incredibly useful.
Real-World Applications of Trigonometry
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