## Math 9 Chapter 1 Lesson 3: Trigonometry table

## 1. Summary of theory

### 1.1. Structure of the trigonometric table

– The trigonometric table includes table VIII, table IX, table X of the book “number table with 4 decimal places” by author VM Bradyz

– People make a table based on the property: If two acute angles \(\alpha\) and \(\beta\) are complementary (\(\alpha +\beta=90^{\circ}\)) then \( sin\alpha =cos\beta ,cos\alpha =sin\beta ,tan\alpha =cot\beta ,cot\alpha =tan\beta\)

– Table VIII is used to calculate the sine and cosine values of acute angles and also to find the angle when knowing the sine and cosine of that angle. Has a structure of 16 columns and rows

– Columns 1 and 13 record integer degrees. Column 1 from top to bottom records the number of degrees increasing from \(0^{\circ}\) to \(90^{\circ}\), column 13 vice versa descending.

– The last three columns record values used to correct for different angles 1′ , 2′ , 3′

– Table IX is used to find the value of tan of angles from \(0^{\circ}\) to \(76^{\circ}\) and cots of angles from \(14^{\circ}\) to \(90^{\circ}\) and vice versa find the angle when tan and cot are known. Structure similar to table VIII

– Table X is used to find the value of tan the angles \(76^{\circ}\) to \(89^{\circ}59{}’\) and cot the angles from \(1{}’\) to \(14^{\circ}\) and vice versa find the acute angle when tan and cot . are known

### 1.2. How to use the table

a) Find the trigonometric ratio of a given acute angle

**Step 1: **Look up the number of degrees in column 1 for sin and tan (column 13 for cos and cots)

**Step 2: **Look up the minutes in row 1 for sin and tan (last row for cos and cot)

**Step 3: **Take the value as delivery with the number of degrees and the column for the number of minutes

In case the minute is not a multiple of 6, we take the minute column with the closest value and see the difference in the correction

b) Find the measure of an acute angle with the trigonometric ratio of that angle: Look up the value of the trigonometric ratio with the appropriate table then align to the degrees and minutes column corresponding to the ratio. We will have the measure of the angle we need to find.

## 2. Illustrated exercise

**Question 1.** Use the trigonometric table to find the following trigonometric ratios: \(sin45^{\circ}12{}’,cos41^{\circ}30{}’\)

**Solution guide**

Look up table VIII in the 4th column and the line corresponding to \(45^{\circ}\) in the first column, we get \(sin45^{\circ}12{}’=0.7096\)

Similarly for the 7th column and the line corresponding to \(41^{\circ}\) in the 13th column we get \(cos41^{\circ}30{}’=0.749\)

**Verse 2. **Use trigonometry to find acute angle x know: \(tanx=3,582\)

**Solution guide**

Look up in Table IX the values closest to 3,582 and we can see that \(x=74^{\circ}24{}’\)

**Verse 3. **Arrange in ascending order the following trigonometric ratios: \(sin78^{\circ}, cos15^{\circ}, sin50^{\circ}, cos80^{\circ}\)

**Solution guide**

With the corners \(0^0<\alpha, \beta<90^0\) then if \(\alpha > \beta \Leftrightarrow {\rm{ }}\sin \alpha > \sin \beta \)

Change: \(\cos 80^{0}=\sin 10^0, \cos 15^{0}=\sin 75^0 \). From there arrange.

**Verse 4. **Compare \(\tan 25^0 \) and \(\sin 25^0 \)

**Solution guide**

Method 1: Look up the table and see \(sin25^{\circ}\approx 0.423<0.466\approx tan25^{\circ}\)

Method 2: we have: \(tan25^{\circ}=\frac{sin25^{\circ}}{cos25^{\circ}}>sin25^{\circ}\) because \(0<\cos 25^0 <1\)

**Question 5. **Use only the trigonometric table to approximate the value of \(tan74^{\circ}8{}’\)

**Solution guide**

First looking at table IX we see that the angle \(74^{\circ}8{}’\) is close to the angle \(74^{\circ}6{}’\) and the error is \(2{}’ \)

continue to look at the correction along the same line as 8. should take the value of \(tan74^{\circ}6{}’=3.511\) plus 0.008

We get \(tan74^{\circ}6{}’=3,519\)

## 3. Practice

### 3.1. Essay exercises

**Question 1.** Use the trigonometric table to find the following trigonometric ratios: \(\sin {{45}^{{}^\circ }}{{12}^{\prime }},\cos {{41}^{{ }^\circ }}{{30}^{\prime }}\)

**Verse 2. **Use trigonometry to find acute angle x know: \(\tan x=3,582\)

**Verse 3. **Arrange in ascending order the following trigonometric ratios: \(\sin {{78}^{{}^\circ }},\cos {{15}^{{}^\circ }},\sin {{50}^{{}^\circ }},\cos {{80}^{{}^\circ }}\)

**Verse 4. **Compare \(\tan {{25}^{{}^\circ }}\) and \(\sin {{25}^{{}^\circ }}\)

**Question 5.** Use only the trigonometric table to approximate the value of \(\tan {{74}^{{}^\circ }}{{8}^{\prime }}\)

### 3.2. Multiple choice exercises

**Question 1. **Use only trigonometric tables to calculate the value of trigonometric ratios: \(tan71^{\circ}48{}’\)

A. 3.024

B. 3.042

C. 3.060

D. 3.078

**Verse 2.** Calculate the measure of apex angle \(x\) knowing that: \(tanx=0,1016\)

A. \(4^{\circ}12{}’\)

B. \(1^{\circ}30{}’\)

C. \(5^{\circ}48{}’\)

D. \(3^{\circ}30{}’\)

**Verse 3. **Value of \(sin49^{\circ}50{}’\)

A. 0.7638

B. 0.7640

C. 0.7642

D. 0.7644

**Verse 4.** Value of \(cos40^{\circ}36{}’\):

A. 0.7604

B. 0.7615

C. 0.7581

D. 0.7593

**Question 5. **Value of \(cot17^{\circ}30{}’\)

A. 3.376

B. 3.191

C. 3.172

D. 3.152

## 4. Conclusion

Through this lesson, you should know the following:

- Know the structure of the trigonometric table.
- Know how to use the trigonometric table.

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