Problem 103 - Algebra and Trigonometry Real Mathematics, Real People

Question

Evaluate the function for the given values of \(t\). Values of \(t\) (a) \(t=3\) (a) \(t=2\) (a) \(t=0.3\) (a) \(t=0.1\) (b) \(t=\pi\) (b) \(t=0\) (b) \(t=-\pi / 4\) (b) \(t=-2 \pi / 3\) Trigonometric Function $$f(t)=\cos t$$

Step-by-Step Solution

Answer

The function evaluations are \(f(3) \approx -0.9900\), \(f(2) \approx -0.4161\), \(f(0.3) \approx 0.9553\), \(f(0.1) \approx 0.9950\), \(f(\pi) = -1\), \(f(0) = 1\), \(f(-\pi / 4) \approx 0.7071\), \(f(-2\pi / 3) = -0.5\).

Step 1: Evaluate for \(t = 3\)

Substitute \(t = 3\) into the function to get \(f(3) = \cos(3)\). Use a calculator to find the cosine of 3, which is approximately -0.9900.

Step 2: Evaluate for \(t = 2\)

Substitute \(t = 2\) into the function to get \(f(2) = \cos(2)\). Use a calculator to find the cosine of 2, which is approximately -0.4161.

Step 3: Evaluate for \(t = 0.3\)

Substitute \(t = 0.3\) into the function to get \(f(0.3) = \cos(0.3)\). Use a calculator to find the cosine of 0.3, which is approximately 0.9553.

Step 4: Evaluate for \(t = 0.1\)

Substitute \(t = 0.1\) into the function to get \(f(0.1) = \cos(0.1)\). Use a calculator to find the cosine of 0.1, which is approximately 0.9950.

Step 5: Evaluate for \(t = \pi\)

Substitute \(t = \pi\) into the function to get \(f(\pi) = \cos(\pi)\). The cosine of \(\pi\) is -1.

Step 6: Evaluate for \(t = 0\)

Substitute \(t = 0\) into the function to get \(f(0) = \cos(0)\). The cosine of 0 is 1.

Step 7: Evaluate for \(t = -\pi / 4\)

Substitute \(t = -\pi / 4\) into the function to get \(f(-\pi / 4) = \cos(-\pi/4)\). The cosine of \(-\pi / 4\) is approximately 0.7071.

Step 8: Evaluate for \(t = -2\pi / 3\)

Substitute \(t = -2\pi / 3\) into the function to get \(f(-2\pi / 3) = \cos(-2\pi / 3)\). The cosine of \(-2\pi / 3\) is approximately -0.5.