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Write the following in terms of a, b and c. (a) cos θ (b) tan θ (c) sin(90° - θ)

Mathematics

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7 parts
3.1

21 learners21 \text{ learners}

3.2
54
3.3
31
3.4

Minimum=3,Q1=16.5,Median=29,Q3=37,Maximum=57\text{Minimum}=3, Q_1=16.5, \text{Median}=29, Q_3=37, \text{Maximum}=57

a

bc\frac{b}{c}

b

ab\frac{a}{b}

c

bc\frac{b}{c}

Step-by-step solution

13 steps

Part 3

The stem and leaf diagram represents the marks out of 60.
The data points, in ascending order, are:
3,7,15,15,16,17,22,24,28,29,29,31,31,31,34,36,38,43,44,44,573, 7, 15, 15, 16, 17, 22, 24, 28, 29, 29, 31, 31, 31, 34, 36, 38, 43, 44, 44, 57.
There are n=21n=21 data points.

Part 3.1

21 learners21 \text{ learners}

How many learners wrote the test?

  1. Count the total number of leaves in the stem and leaf diagram.

    Number of leaves = 2+4+5+6+3+1=212+4+5+6+3+1 = 21.

    21 learners\boxed{21 \text{ learners}}

Part 3.254

Calculate the range.

  1. Identify the maximum and minimum values from the data.

    Maximum value = 57.
    Minimum value = 3.

  2. Calculate the range.

    Range=Maximum value−Minimum value\text{Range} = \text{Maximum value} - \text{Minimum value} Range=57−3\text{Range} = 57 - 3 Range=54\text{Range} = \boxed{54}

Part 3.331

Determine the mode.

  1. Identify the value(s) that appear most frequently in the data set.

    The value 31 appears three times, which is more than any other value.

    Mode=31\text{Mode} = \boxed{31}

Part 3.4

Minimum=3,Q1=16.5,Median=29,Q3=37,Maximum=57\text{Minimum}=3, Q_1=16.5, \text{Median}=29, Q_3=37, \text{Maximum}=57

Determine the five number summary.

  1. Identify the minimum and maximum values.

    Minimum value = 3.
    Maximum value = 57.

  2. Calculate the median (Q2Q_2).

    For n=21n=21 (odd), the median is the (n+12)th(\frac{n+1}{2})^{th} value.

    Q2=(21+12)th value=11th valueQ_2 = \left(\frac{21+1}{2}\right)^{th} \text{ value} = 11^{th} \text{ value}

    The 11th11^{th} value in the sorted list is 29.

    Q2=29Q_2 = 29
  3. Calculate the lower quartile (Q1Q_1).

    Q1Q_1 is at the 14(n+1)th\frac{1}{4}(n+1)^{th} position.

    Q1=14(21+1)th position=5.5th positionQ_1 = \frac{1}{4}(21+1)^{th} \text{ position} = 5.5^{th} \text{ position}

    Q1Q_1 is the average of the 5th5^{th} and 6th6^{th} values: 16+172\frac{16+17}{2}.

    Q1=332=16.5Q_1 = \frac{33}{2} = 16.5
  4. Calculate the upper quartile (Q3Q_3).

    Q3Q_3 is at the 34(n+1)th\frac{3}{4}(n+1)^{th} position.

    Q3=34(21+1)th position=16.5th positionQ_3 = \frac{3}{4}(21+1)^{th} \text{ position} = 16.5^{th} \text{ position}

    Q3Q_3 is the average of the 16th16^{th} and 17th17^{th} values: 36+382\frac{36+38}{2}.

    Q3=742=37Q_3 = \frac{74}{2} = 37
  5. State the five-number summary.

    Minimum=3,Q1=16.5,Median=29,Q3=37,Maximum=57\boxed{\text{Minimum}=3, Q_1=16.5, \text{Median}=29, Q_3=37, \text{Maximum}=57}

Part 4

From the right-angled triangle diagram:

  • The side opposite to angle θ\theta is aa.
  • The side adjacent to angle θ\theta is bb.
  • The hypotenuse is cc.

Part 4.1

1 Write the following in terms of aa, bb, and cc.

Part a

bc\frac{b}{c}

cos⁡θ\cos \theta

  1. Use the definition cos⁡θ=AdjacentHypotenuse\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}.

    cos⁡θ=bc\cos \theta = \boxed{\frac{b}{c}}

Part b

ab\frac{a}{b}

tan⁡θ\tan \theta

  1. Use the definition tan⁡θ=OppositeAdjacent\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}.

    tan⁡θ=ab\tan \theta = \boxed{\frac{a}{b}}

Part c

bc\frac{b}{c}

sin⁡(90∘−θ)\sin(90^{\circ} - \theta)

  1. Use the trigonometric identity sin⁡(90∘−θ)=cos⁡θ\sin(90^{\circ} - \theta) = \cos \theta.

  2. Substitute the expression for cos⁡θ\cos \theta from part (a).

    sin⁡(90∘−θ)=cos⁡θ=bc\sin(90^{\circ} - \theta) = \cos \theta = \boxed{\frac{b}{c}}
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