Chapter 4.2, Problem 73E

Solution

To identify: Which column gives sine values, which gives cosine values, and which gives tangent values without using a calculator. Below is a simplified trig table for angles between 40° and 50°.

  $\begin{array}{cccc}Angle& ?& ?& ?\\ 40°& 0.8391& 0.6428& 0.7660\\ 42°& 0.9004& 0.6691& 0.7431\\ 44°& 0.9657& 0.6947& 0.7193\\ 46°& 1.0355& 0.7193& 0.6947\\ 48°& 1.1106& 0.7431& 0.6691\\ 50°& 1.1918& 0.7660& 0.6428\end{array}$

Column 1 represents tangent, column 2 represents sine and column 3 represents cosine.

Given information:

The given table is:

  $\begin{array}{cccc}Angle& ?& ?& ?\\ 40°& 0.8391& 0.6428& 0.7660\\ 42°& 0.9004& 0.6691& 0.7431\\ 44°& 0.9657& 0.6947& 0.7193\\ 46°& 1.0355& 0.7193& 0.6947\\ 48°& 1.1106& 0.7431& 0.6691\\ 50°& 1.1918& 0.7660& 0.6428\end{array}$

Explanation:

Consider the given information.

  $\begin{array}{l}\text{Column }\left(\text{?}\right)\text{1 = tangent}\hfill \\ \text{Column }\left(\text{?}\right)\text{2 = sine}\hfill \\ \text{Column }\left(\text{?}\right)\text{3 = cosine}\hfill \end{array}$

The first two columns have numbers that increase as the angle increases. This is because when making the angle bigger, the opposite side will also get bigger.

  

The first column represents the tangent because the value of cosine and sine both lies between $-1$ to 1.

Since the first column has value greater than 1 so it must be tangent.

The value of $\sin 0°=0$ and the value of $\sin 90°=1$ . Thus, from 0 to $90°$ the value of sine function increases.

Here it can be observed that this is happing in column 2.

Also, the value of $\cos 0°=1$ and the value of $\cos 90°=0$ . Thus, from 0 to $90°$ the value of cosine function decreases.

Here it can be observed that this is happing in column 3.

With cosine, as the angle gets bigger, the adjacent side gets smaller. This makes sense if you picture stretching the triangle up and down. The opposite side will get bigger, but the adjacent will get smaller.

Thus, Column 1 represents tangent, column 2 represents sine and column 3 represents cosine.