Ask a New Question
Menu

Show that cos x+π = -cos⁡(x)

Ambiguous Case of the Law of Sines (3 pts)
Triangle DEF is given where m<D=40°, d=20 cm, and the height h=12 cm. How many distinct triangles can be made with the given measurements? Explain your answer fully.
Solve the Trigonometric Equation (3pts)
Solve the equation 4sin2x=1, for 0 ≤ x ≤ 2π.
brief explanations only!

1 answer

Showing that \( \cos(x + \pi) = -\cos(x) \)

Using the cosine addition formula:

\[ \cos(x + \pi) = \cos x \cos \pi - \sin x \sin \pi \]

We know that:

\(\cos \pi = -1\) and \(\sin \pi = 0\), so:

\[ \cos(x + \pi) = \cos x (-1) - \sin x (0) = -\cos x \]

Thus, \( \cos(x + \pi) = -\cos x \).

Ambiguous Case of the Law of Sines (Triangle DEF)

Given:

  • \( m<D = 40° \)
  • \( d = 20 , \text{cm} \)
  • Height \( h = 12 , \text{cm} \)

First, calculate the angle at \( D \):

Using \( h = d \sin D \):

\[ 12 = 20 \sin(40°) \]

Calculate \( \sin(40°) \):

\[ \sin(40°) \approx 0.6428 \quad \Rightarrow \quad 20 \cdot 0.6428 \approx 12.856 , \text{cm} \]

Since \( 12 < 20 \sin(40°) \), there are two possible triangles (one acute, one obtuse). Thus, two distinct triangles can be formed.

Solve the Trigonometric Equation \( 4\sin(2x) = 1 \)

  1. Rewrite the equation:

    \[ \sin(2x) = \frac{1}{4} \]

  2. Solve for \( 2x \):

    The general solutions for \( \sin(\theta) = k \) are:

    \[ 2x = \arcsin\left(\frac{1}{4}\right) + 2k\pi \quad \text{and} \quad 2x = \pi - \arcsin\left(\frac{1}{4}\right) + 2k\pi \]

  3. Calculate \( \arcsin\left(\frac{1}{4}\right) \):

    \( \arcsin\left(\frac{1}{4}\right) \approx 0.2527 \) radians.

  4. Finding \( x \):

    Divide by 2:

    \[ x = \frac{0.2527}{2} + k\pi \quad \text{and} \quad x = \frac{\pi - 0.2527}{2} + k\pi \]

  5. Thus, valid solutions for \( 0 \leq x \leq 2\pi \) occur when \( k = 0 \):

    • \( x_1 = \frac{0.2527}{2} \)
    • \( x_2 = \frac{\pi - 0.2527}{2} \)

Calculating these:

\[ x_1 \approx 0.1264, \quad x_2 \approx 1.0694 , \text{(approximately)} \]

And repeating for \( k = 1 \):

  • \( x_3, x_4 ...\) would yield values exceeding \( 2\pi \).

Final answers: \( x \approx 0.1264 \) and \( x \approx 1.0694 \) (two solutions).