Evaluate cos(theta)=15/17

Solution:

Step 1:

Apply the inverse cosine function to both sides to isolate $\theta$: $\theta ={\cos }^{-1}\left(\frac{15}{17}\right)$

Step 2:

Calculate the value on the right-hand side.

Step 2.1:

Compute ${\cos }^{-1}\left(\frac{15}{17}\right)$ to find $\theta$: $\theta \approx 0.48995732$

Step 3:

Since cosine is positive in both the first and fourth quadrants, determine the second angle by subtracting the principal angle from $2\pi$: $\theta =2\pi -0.48995732$

Step 4:

Compute the value of $\theta$.

Step 4.1:

Expand the expression: $\theta =2\pi -0.48995732$

Step 4.2:

Simplify the expression $2\pi -0.48995732$.

Step 4.2.1:

Multiply $2$ by $\pi$: $\theta =6.2831853-0.48995732$

Step 4.2.2:

Subtract $0.48995732$ from $6.2831853$: $\theta \approx 5.79322798$

Step 5:

Identify the period of the cosine function.

Step 5.1:

Calculate the period using the formula $\frac{2\pi }{|b|}$.

Step 5.2:

Since $b=1$ for the cosine function, replace $b$ with $1$: $\frac{2\pi }{|1|}$

Step 5.3:

The absolute value of $1$ is $1$: $\frac{2\pi }{1}$

Step 5.4:

Divide $2\pi$ by $1$ to get the period: $2\pi$

Step 6:

The period of cosine is $2\pi$, so the function repeats every $2\pi$ radians. Thus, $\theta =0.48995732+2\pi n$ and $\theta =5.79322798+2\pi n$ for any integer $n$.

Step 7:

Present the result in different formats.

Exact Form: $\cos (\theta )=\frac{15}{17}$

Decimal Form: $\cos (\theta )\approx 0.88235294\dots$

Knowledge Notes:

1. Inverse Cosine Function: The inverse cosine function, denoted as ${\cos }^{-1}$ or $arccos$, is used to find the angle whose cosine is a given number. It is the inverse operation of the cosine function.

2. Cosine Function: The cosine function relates the angle of a right-angled triangle to the ratio of the adjacent side over the hypotenuse. It is positive in the first and fourth quadrants of the unit circle.

3. Radians: Radians are a unit of angular measure used in mathematics. One full revolution (360 degrees) is equal to $2\pi$ radians.

4. Period of Cosine Function: The period of the cosine function is the length of the interval on which the function completes one full cycle. For the standard cosine function, the period is $2\pi$ radians.

5. General Solution for Trigonometric Equations: The general solution for trigonometric equations like $\cos (\theta )=\frac{15}{17}$ takes into account that trigonometric functions are periodic. The solutions are given by $\theta ={\cos }^{-1}\left(\frac{15}{17}\right)+2\pi n$ and $\theta =2\pi -{\cos }^{-1}\left(\frac{15}{17}\right)+2\pi n$, where $n$ is any integer.

6. Exact vs. Decimal Form: Trigonometric values can be expressed in exact form (using fractions and square roots) or in decimal form (an approximation). Exact form is more precise, while decimal form is often used for practical calculations.