Problem 68 - Algebra and Trigonometry

Question

Find the exact value of each expression. Do not use a calculator. $$ \cos 12^{\circ} \sin 78^{\circ}+\cos 78^{\circ} \sin 12^{\circ} $$

Step-by-Step Solution

Answer

The exact value of the expression is 1.

Step 1: Recognize the identity

Recognize that the expression is in the form of the sine of the sum of two angles identity: \(\sin(a + b) = \sin a \cos b + \cos a \sin b\). In this case, \(a = 12^{\circ}\) and \(b = 78^{\circ}\).

Step 2: Apply the identity

Substitute the values of \(a\) and \(b\) into the identity. This gives \(\sin(12^{\circ} + 78^{\circ})\).

Step 3: Calculate the sum

Add the degrees together to get \(90^{\circ}\). So, our expression simplifies to \(\sin(90^{\circ})\).

Step 4: Find the exact value

Using the unit circle or the knowledge of trigonometric values, one can see that \(\sin(90^{\circ}) = 1\) as the y-coordinate at that point on the unit circle is 1.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in mathematics that provide relationships between the trigonometric functions. One such crucial identity is the sine sum of two angles formula: \[\begin{equation}\sin(a + b) = \sin a \cos b + \cos a \sin b\end{equation}\]Understanding this identity is not just about memorizing the formula; it's about recognizing patterns that often simplify complex trigonometric expressions. As in our exercise, when faced with a combination of sine and cosine functions, you can often reconfigure them into a single function by noticing they fit the structure of the sine sum identity. By rearranging the expression to match this identity, as shown in the problem's step-by-step solution, the simplification process becomes more manageable and intuitive.

Exact Trigonometric Values

When solving trigonometric problems, being familiar with the exact values of trigonometric functions at special angles can save a great deal of time and effort. These special angles, often found at multiples of 30 and 45 degrees, have sine, cosine, and tangent values that can be exactly determined without a calculator. For instance, as seen in the exercise solution, \[\begin{equation}\sin(90^{\textdegree})\end{equation}\]is one of these angles, having an exact value of 1. This knowledge is incredibly useful and can instantly transform a complex-looking expression into a much simpler form. It's essential to learn these by heart or know how to quickly derive them from the unit circle to efficiently solve trigonometry problems.

Unit Circle

The unit circle is a powerful concept that represents all the possible angles and the corresponding values of sine, cosine, and tangent around a circle of radius 1 centered at the origin of a coordinate system. The beauty of the unit circle lies in its ability to provide a geometric interpretation of the trigonometric functions. For example, the sine of an angle corresponds to the y-coordinate of the point where the angle's terminal side intersects the unit circle. In the context of our problem, knowing that the sin function reaches its maximum value of 1 at a 90-degree angle allows us to use the unit circle to instantly find that \[\begin{equation}\sin(90^{\textdegree}) = 1\end{equation}\]As learners progress in trigonometry, familiarity with the unit circle becomes indispensable for understanding and visualizing how trigonometric functions behave for different angles and how identities play out geometrically.