7.1 Solving Trigonometric Equations with Identities

  1. Given \(f(x)=x^2+x\), solve \(f(x)=0\).

  2. Given \(f(x)= \sin^2(x) + \sin(x)\), solve \(f(x)=0\).

  3. Solve \(2\sin^2(x) + \sin(x)=0\) for all x, \(0 \leq x < 2\pi\).

  4. Solve \(3\sec^2(x) - 5 \sec(x) - 2 = 0\) for all x, \(0 \leq x < 2\pi\).

Below is a reminder of all the identities we have discussed so far.

  1. \(12\sin^2(x) - 13\cos(x)-15 = 0\)
  2. \(10\sin(x)\cos(x)=6\cos(x)\)
  3. \(\sec(2x)=3\)
  4. \(\sec(x)\sin(x)-2\sin(x)=0\)
  5. \(2\sin^2(x)+3\sin(x)+1=0\)
  6. \(\tan^3(x)=4\tan(x)\)
  7. \(\tan(x)-3\sin(x)=0\)

7.2 Addition and Subtraction Identities

  1. Prove the sum angle identities below.

  1. Find the exact value of \(\cos(75^\circ)\).

  2. Find the exact value of \(\sin(\frac{\pi}{12})\).

  3. Solve \(\sin(x)\sin(2x)+\cos(x)\cos(2x)=\frac{\sqrt{3}}{2}\)

  4. Rewrite \(\sin(x-\frac{\pi}{4})\) in terms of \(\sin(x)\) and \(\cos(x)\).

  5. Rewrite \(4\cos(x-\frac{\pi}{3})\) in terms of \(\sin(x)\) and \(\cos(x)\).

  6. Prove the first product-to-sum identity below:

  7. Write \(\sin(2x)\sin(4x)\) as a sum or difference.

  8. Evaluate \(\cos(\frac{11\pi}{12})\cos(\frac{\pi}{12})\).

  1. Evaluate \(\cos(15^\circ)-\cos(75^\circ)\).
  2. Prove the identity: \[\frac{\cos(4t)-cos(2t)}{\sin(4t)+\sin(2t)}=-\tan(t)\]
  3. Solve: \[\sin(\pi t)+\sin(3\pi t)=\cos(\pi t)\]

  1. Rewrite \(4\sqrt{3} \sin(2x) - 4\cos(2x)\) as a single sinusoidal function using technique described above.
  2. Solve \(4\sqrt{3} \sin(2x) - 4\cos(2x)=0\).

7.3 Double Angle Identities

  1. Prove the double angle identities.

  1. If \(\sin(x)=\frac{3}{5}\) find the exact values of \(\sin(2x)\) and \(\cos(2x)\).

  2. Simplify \(8\sin(3x)\cos(3x)\).

  3. Simplify \(\frac{\cos(2x)}{\cos(x)-\sin(x)}\).

  4. Prove \(\sec(2x)=\frac{\sec^2(x)}{2-\sec^2(x)}\).

  5. Solve \(\cos(2x)=\cos(x)\) for all \(0 \leq x < 2\pi\).

  6. Use a double angle identity to prove the power reduction formulas:

\[\cos^2(x)=\frac{\cos(2x)+1}{2}\]

\[\sin^2(x)=\frac{1-\cos(2x)}{2}\]

  1. Use an identity to find the exact value of \(\cos^2(75^\circ)-\sin^2(75^\circ)\).

  2. Prove the half-angle formulas:

  1. Rewrite \(\cos^4(x)\) without any powers.
  2. Find an exact value for \(\sin(15^\circ)\).
  3. Find an exact value for \(\cos(112.5^\circ)\).

7.4 Modeling Changing Amplitude and Midline

  1. A population of elk currently averages 2000 elk, and that average has been growing by 80 elk each year. Due to seasonal fluctuation, the population oscillates from 50 below average in the winter up to 50 above average in the summer. Find a function that models the number of elk after t years, starting in the winter.

  2. A population of elk currently averages 2000 elk, and that average has been growing by 4% each year. Due to seasonal fluctuation, the population oscillates from 50 below average in the winter up to 50 above average in the summer. Find a function that models the number of elk after t years, starting in the winter.

  1. The number of tourists visiting a ski and hiking resort averages 4000 people annually and oscillates seasonally, 1000 above and below the average. Due to a marketing campaign, the average number of tourists has been increasing by 200 each year. Write an equation for the number of tourists after t years, beginning at the peak season.

  2. Given the function \(g(x)=(x^2-1)+8\cos(x)\), describe the midline and amplitude using words.

  1. A spring with natural length of feet inches is pulled back 6 feet and released. It oscillates once every 2 seconds. Its amplitude decreases by 20% each second. Find a function that models the position of the spring t seconds after being released.

  2. A spring with natural length of 30 cm is pulled out 10 cm and released. It oscillates 4 times per second. After 2 seconds, the amplitude has decreased to 5 cm. Find a function that models the position of the spring.

  3. In AM (Amplitude Modulated) radio, a carrier wave with a high frequency is used to transmit music or other signals by applying the to-be-transmitted signal as the amplitude of the carrier signal. A musical note with frequency 110 Hz (Hertz = cycles per second) is to be carried on a wave with frequency of 2 KHz (KiloHertz = thousands of cycles per second). If the musical wave has an amplitude of 3, write a function describing the broadcast wave.

  1. A certain stock started at a high value of $7 per share, oscillating monthly above and below the average value, with the oscillation decreasing by 2% per year. However, the average value started at $4 per share and has grown linearly by 50 cents per year.
    1. Find a formula for the midline and the amplitude.
    2. Find a function \(S(t)\) that models the value of the stock after t years.