Problem 54 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$67^{\circ} 30^{\prime}$$

Step-by-Step Solution

Answer

The exact values of sine, cosine, and tangent of the angle \(67^{\circ} 30^{\prime}\) are approximately 0.8284, 0.5590, and 1.4801, respectively.

Step 1: Converting Angle

First, convert the mixed number angle \(67^{\circ} 30^{\prime}\) into decimal angle. We know that \( 1^{\circ}= 60^{\prime}\), so \(30^{\prime} = (30/60) = 0.5^{\circ}\). Therefore, the given angle in decimal is \(67.5^{\circ}\).

Step 2: Applying Half-angle Formulas

Now, apply the half-angle formulas. Since \( \theta/2 \) for the given problem \( \theta= 67.5^{\circ} \) falls in the first quadrant (where all trigonometric functions are positive), we will consider the positive values.\[ \sin(\theta/2)= \sqrt{\frac{1-\cos67.5^{\circ}}{2}} \]\[ \cos(\theta/2)= \sqrt{\frac{1+\cos67.5^{\circ}}{2}} \]\[ \tan(\theta/2)= \sqrt{\frac{1-\cos67.5^{\circ}}{1+\cos67.5^{\circ}}} \]We use known values of cosine from a trigonometric table (or calculator) to solve the above equations.

Step 3: Calculation

Now we calculate the values.For sine function:First, we calculate \( \cos 67.5^{\circ} \) approximately equals to 0.3907.\[ \sin(\theta/2)= \sqrt{\frac{1-0.3907}{2}}=0.8284 \]For cosine function:\[ \cos(\theta/2)= \sqrt{\frac{1+0.3907}{2}}=0.5590 \]For tangent function:\[ \tan(\theta/2)= \sqrt{\frac{1-0.3907}{1+0.3907}}=1.4801 \]