Problem 53 - Algebra and Trigonometry

Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the exact value of \(5 \cos 60^{\circ}+2 \tan \frac{\pi}{4} .\) Do not use a calculator.

Step-by-Step Solution

Answer

The exact value is \frac{9}{2}.

Step 1 - Identify the trigonometric values

Find the exact values of the trigonometric functions given: - \(\cos 60^{\circ} = \frac{1}{2}\) - \(\tan \frac{\pi}{4} = 1\)

Step 2 - Substitute trigonometric values into the expression

Substitute the known values into the expression: \(5 \cos 60^{\circ} + 2 \tan \frac{\pi}{4} = 5 \left(\frac{1}{2}\right) + 2(1)\)

Step 3 - Perform the multiplication

Multiply the constants with their respective trigonometric values: \(5 \cdot \frac{1}{2} = \frac{5}{2}\) \(2 \cdot 1 = 2\)

Step 4 - Add the results together

Add the results from the previous step: \(\frac{5}{2} + 2 = \frac{5}{2} + \frac{4}{2} = \frac{9}{2}\)

Key Concepts

cosine

The cosine function, often written as \(\text{cos}\), is one of the primary trigonometric functions. It relates the angle of a right triangle to the ratio of the adjacent side to the hypotenuse. For instance, \(\text{cos} \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\).\r\rIn the unit circle, the cosine of an angle equals the x-coordinate of the corresponding point. For example:\r

\r
• \(\text{cos} \ 0^{\text{o}} = 1\)
\r
• \(\text{cos} \ 90^{\text{o}} = 0\)
\r
• \(\text{cos} \ 60^{\text{o}} = \frac{1}{2}\)
\r

\rWhen solving problems, it helps to remember these exact values. In our example, \(\text{cos} \ 60^{\text{o}}\) is used. Knowing \(\text{cos} \ 60^{\text{o}} = \frac{1}{2}\) allowed us to substitute easily into our given expression.

tangent

The tangent function, or \(\text{tan}\), is another fundamental trigonometric function. It relates the angle in a right triangle to the ratio of the opposite side to the adjacent side: \(\text{tan} \theta = \frac{\text{Opposite}}{\text{Adjacent}}\).\r\rLike other trigonometric functions, it can be represented using the unit circle. Here are some exact values you should remember:\r

\r
• \(\text{tan} \ 0^{\text{o}} = 0\)
\r
• \(\text{tan} \ 45^{\text{o}} = 1\)
\r
• \(\text{tan} \ 90^{\text{o}} = \text{undefined}\)
\r

\rIn our exercise, we used \(\text{tan} \ \frac{\text{\text{\text{\text{\text{\text{\text{\text{\text{π}}}}}}}}}}{4}\) because \(\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{π}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} = 45^{\text{o}}\) so \(\text{tan} \ \frac{\text{π}}{4} = 1\). This value made our calculations straightforward as we knew exactly what to substitute.

exact values

Exact values in trigonometry are specific, well-known values of trigonometric functions at certain angles. These values often appear in problems requiring you to find solutions without a calculator.\r
\r
\rSome commonly known exact values include:\r

\r
• \(\text{sin} \ 30^{\text{o}} = \frac{1}{2}\)
\r
• \(\text{cos} \ 45^{\text{o}} = \frac{\text{\text{\text{\text{\text{\text{\text{1}}}}}}}}}{\text{\text{\text{\text{\text{\text{\text{√}}}}}}}}{2}}\)
\r
• \(\text{tan} \ 60^{\text{o}} = \text{√}{3}\)
\r

\rIn the given exercise, knowing the exact values for \(\text{cos} \ 60^{\text{o}}\) and \(\text{tan} \ \frac{\text{\text{\text{\text{\text{\text{\text{\text{\text{π}}}}}}}}}}}{4}\) was essential. These allowed us to substitute them directly into the expression and compute without any approximation.\r
\r
\rExact values simplify your calculations and help in solving problems accurately. Always remember to memorize key trigonometric exact values as they will aid you in exams and homework.