Question

If \(\tan \theta=4,\) find the exact value of \(\tan \theta+\tan \left(\frac{\pi}{2}-\theta\right)\)

Short answer

\tan \theta + \tan \big(\frac{\text{\textpi}}{2} - \theta\big) = \frac{17}{4}.

Step-by-step solution

Identify Given Values

It is given that \(\tan \theta = 4\).

Recall the Trigonometric Identity

Recall the co-function identity: \(\tan\big(\frac{\text{\textpi}}{2} - x\big) = \frac{1}{\tan x}\).

Apply the Identity

Using the identity \(\tan \big(\frac{\text{\textpi}}{2} - \theta\big) = \frac{1}{\tan \theta}\) and the given value \(\tan \theta = 4\), we find: \(\tan \big(\frac{\text{\textpi}}{2} - \theta\big) = \frac{1}{4}\).

Add the Tangents

Add the two tangent values: \(\tan \theta + \tan \big(\frac{\text{\textpi}}{2} - \theta\big) = 4 + \frac{1}{4}\).

Simplify the Expression

Combine the values: \(\frac{16}{4} + \frac{1}{4} = \frac{17}{4}\).

Key concepts

tangent function

The tangent function, often abbreviated as 'tan,' is a fundamental trigonometric function. It represents the ratio of the opposite side to the adjacent side in a right-angled triangle. For an angle \( \theta \), it can be defined as follows:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \] In the unit circle, the tangent of an angle is the length of the segment from the origin to the point where the terminal side of the angle intersects the tangent line to the unit circle. This function is periodic, meaning it repeats its values in regular intervals of \( \pi \). Therefore, understanding the tangent function's properties is essential for solving many trigonometric problems.

co-function identities

Co-function identities are some of the most helpful tools in trigonometry. They relate trigonometric functions of complementary angles (angles that add up to \( \frac{\pi}{2} \) or 90 degrees). The identity used in the exercise is:

\[ \tan \left(\frac{\pi}{2} - x\right) = \frac{1}{\tan x} \] This means that the tangent of an angle's complement is the reciprocal of the tangent of the angle itself. In other words, if you know \( \tan \theta \), you can easily find \( \tan \left(\frac{\pi}{2} - \theta\right) \) by taking the reciprocal of \( \tan \theta \). This particular co-function identity is handy when dealing with angles that sum to 90 degrees. In our case, since \( \tan \theta = 4 \), it implies that \( \tan \left(\frac{\pi}{2} - \theta\right) = \frac{1}{4} \). These identities simplify the process of solving various trigonometric problems by transforming complicated expressions into simpler forms.

trigonometric simplification

Trigonometric simplification often involves reducing complex expressions into simpler forms. This can involve combining like terms, using identities, or converting expressions using known values. In our exercise, we simplified by:
1. Recognizing that \( \tan \theta = 4 \).
2. Applying the co-function identity to find \( \tan \left(\frac{\pi}{2} - \theta\right) = \frac{1}{4} \).
3. Adding these two results:\[ \tan \theta + \tan \left(\frac{\pi}{2} - \theta\right) = 4 + \frac{1}{4} \]
4. Combining the fractions, we get:
\[ \frac{16}{4} + \frac{1}{4} = \frac{17}{4} \]
This approach helps in breaking down complex trigonometric problems into more manageable steps and finding exact values more efficiently.