Determine (61.45°) using cosine tables, given that row 61° under column 0.4 gives 0.4787 and SUBTRACT column 5 gives 8.

Grade 10 · Core Mathematics · Lesson 2 of 11

Finding Cosine and Tangent Values of Acute Angles

A Grade 10 Core Mathematics CBE lesson from the sub-strand Trigonometry 1. In it, learners locate acute angles in cosine tables and read four-figure cosine values for whole and decimal degrees.

Strand Measurements and GeometrySub-strand Trigonometry 1Lesson code 10CM-24-L240 minutes · 4 slides

KICD learning outcome (a)

Determine the trigonometric ratios of acute angles from mathematical tables and calculators.

By the end of this lesson, learners should be able to

Locate acute angles in cosine tables and read four-figure cosine values for whole and decimal degrees.
Apply mean difference subtraction correctly when evaluating cosines of angles given to two decimal places.
Locate acute angles in tangent tables and apply mean difference addition for two-decimal-place angles.

Key vocabulary

Cosine table: A mathematical grid giving cosine values of acute angles up to four decimal places.; We look up row 51° in the cosine table to find initial ratio values.
Subtract column: The difference columns in cosine tables used to subtract mean differences for hundredths.; We locate column 4 in the subtract section to adjust the cosine ratio.
Mean difference: A tabular adjustment representing hundredths of a degree added or subtracted from readings.; A mean difference of 8 is subtracted when evaluating cos 61.45° as 0.4779.
Natural tangent: The trigonometric ratio comparing opposite side length to adjacent side length.; The natural tangent of 36.4° is read from tangent tables as 0.7373.

Lesson preview

Slide 1 · Introduction to Cosine TablesOpening · 10 min

Figure 1: A segment of a natural cosines table showing how to find the cosine of 51.34 degrees.

To evaluate an angle such as $\cos ({51.34}^{\circ})$ , consider how cosine tables are arranged. Locate row $51^{\circ}$ under column $x$ and move across to column $0.3$ . Note the four-figure number at this primary intersection. Next, move along the same row to column $4$ under the SUBTRACT section for the mean difference. Subtract this value from the final digit position of the main reading to get the complete four-figure cosine ratio. Use Figure 1 to follow this process for finding the cosine of 51.34 degrees.

Cosine decreases as acute angles increase, so mean differences must always be subtracted.
Locate whole-degree angles down column x and first decimal digits along top main columns.
Read the main intersection value before checking the hundredths digit in difference columns.
Subtract the mean difference digit from the final position of the main table reading.

Worked example. Locate row

54^{\circ}

down column

x

and move across to column

0.6

to read

0.5793

. Move along row

54^{\circ}

to difference column

3

under SUBTRACT to find mean difference

4

. Subtract

0.0004

from

0.5793

to obtain

0.5789

. Answer: $\cos ({54.63}^{\circ}) = 0.5789$ .

Check question: Determine $\cos ({61.45}^{\circ})$ using cosine tables, given that row $61^{\circ}$ under column $0.4$ gives $0.4787$ and SUBTRACT column $5$ gives $8$ .

0.4779
0.4795
0.4707
0.3987

Show answer

0.4779. The value from the main intersection is $0.4787$ . The mean difference of $8$ means $0.0008$ . Since cosine values decrease as the angle increases, we subtract the mean difference: $0.4787 - 0.0008 = 0.4779$ .

Slide 2 · Modelling Cosine Table Lookups with Decimal AnglesModel · 10 min

Figure 2: A diagram illustrating the steps to find the cosine of 52.38 degrees using a mathematical table.

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All 4 slides with diagrams and class activities
Teacher notes, SNE adaptations and assessment checks
Homework and the other 10 lessons in this sub-strand

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Finding Cosine and Tangent Values of Acute Angles

By the end of this lesson, learners should be able to

Key vocabulary

Lesson preview

Teach the full lesson on the projector

All 11 lessons in Trigonometry 1