Question

If \(\tan\theta = \frac{5}{12}\), then find the value of \(\sin\theta + \cos\theta\).

Hint:

Memorizing common Pythagorean triplets like (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25) will significantly speed up solving trigonometry problems.

Answer 1

Step 1: Understanding the Concept:
Given the tangent of an angle, we can find the sine and cosine of that angle by constructing a right-angled triangle, finding all its sides using the Pythagorean theorem, and then applying the definitions of sine and cosine.

Step 2: Key Formula or Approach:
We are given \(\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}\).
We will find the Hypotenuse using \((\text{Hypotenuse})^2 = (\text{Opposite})^2 + (\text{Adjacent})^2\).
Then we calculate \(\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\) and \(\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\).

Step 3: Detailed Explanation:
We are given \(\tan\theta = \frac{5}{12}\).
In a right-angled triangle, let:
Opposite side = 5
Adjacent side = 12
Let the hypotenuse be 'h'. By the Pythagorean theorem: \[ h^2 = 5^2 + 12^2 \] \[ h^2 = 25 + 144 = 169 \] \[ h = \sqrt{169} = 13 \] So, the Hypotenuse = 13.
Now we find \(\sin\theta\) and \(\cos\theta\): \[ \sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{5}{13} \] \[ \cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{12}{13} \] Finally, we calculate the value of \(\sin\theta + \cos\theta\): \[ \sin\theta + \cos\theta = \frac{5}{13} + \frac{12}{13} = \frac{5+12}{13} = \frac{17}{13} \]

Step 4: Final Answer:
The value of \(\sin\theta + \cos\theta\) is \(\frac{17}{13}\).