If \sin \theta = \frac{4}{7}, what is \cos \theta
The Correct Answer and Explanation is:
Of course. Here is the correct answer and a detailed explanation.
Answer:
cos θ = √33 / 7
Explanation
To find the value of cosine theta (cos θ) when you know the value of sine theta (sin θ), you can use two primary methods, both of which rely on the properties of right-angled triangles.
Method 1: Using SOH CAH TOA and the Pythagorean Theorem
The acronym SOH CAH TOA helps us remember the trigonometric ratios in a right-angled triangle:
- SOH: Sin θ = Opposite / Hypotenuse
- CAH: Cos θ = Adjacent / Hypotenuse
- TOA: Tan θ = Opposite / Adjacent
We are given that sin θ = 4/7. Based on the SOH definition, we can imagine a right-angled triangle where the side opposite angle θ has a length of 4, and the hypotenuse has a length of 7.
To find cos θ, we need the length of the adjacent side. We can find this using the Pythagorean theorem, which states that for a right-angled triangle, a² + b² = c², where a and b are the two shorter sides (legs) and c is the hypotenuse.
Let the adjacent side be ‘a’.
a² + (Opposite)² = (Hypotenuse)²
a² + 4² = 7²
a² + 16 = 49
Now, we solve for a²:
a² = 49 – 16
a² = 33
To find the length ‘a’, we take the square root:
a = √33
Now that we know the adjacent side (√33) and the hypotenuse (7), we can find cos θ using the CAH definition:
cos θ = Adjacent / Hypotenuse = √33 / 7
Method 2: Using the Pythagorean Identity
A fundamental identity in trigonometry is sin²θ + cos²θ = 1. This equation is true for any angle θ. We can use this to solve for cos θ directly.
We are given sin θ = 4/7. Let’s substitute this value into the identity:
(4/7)² + cos²θ = 1
16/49 + cos²θ = 1
Now, isolate cos²θ by subtracting 16/49 from both sides:
cos²θ = 1 – 16/49
cos²θ = 49/49 – 16/49
cos²θ = 33/49
Finally, take the square root of both sides to find cos θ:
cos θ = √(33/49)
cos θ = √33 / 7
Both methods yield the same result. The answer is already simplified, as 33 has no perfect square factors.
