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AQA GCSE · Question 25 · Geometry and Measures
Show that the value of 6 sin 30° + 2 cos 30° × 4 tan 30° is an integer.
Show that the value of 6 sin 30° + 2 cos 30° × 4 tan 30° is an integer.
How to approach this question
1. Recall the exact trigonometric values for 30°. You can derive these from an equilateral triangle of side length 2, split in half.
- sin 30°
- cos 30°
- tan 30°
2. Substitute these exact values (which will involve fractions and surds) into the given expression.
3. Follow the order of operations (BIDMAS/PEMDAS). This means you must do the multiplication part (2 cos 30° × 4 tan 30°) before the addition.
4. Simplify the expression. The surds (√3) should cancel out.
5. Calculate the final numerical value and state that it is an integer.
Full Answer
We need to use the exact values for the trigonometric functions of 30°.
sin 30° = 1/2
cos 30° = √3 / 2
tan 30° = 1 / √3
Now substitute these values into the expression:
Expression = 6(1/2) + 2(√3 / 2) × 4(1 / √3)
Step 1: Evaluate the first term.
6(1/2) = 3
Step 2: Evaluate the second part (the multiplication).
2(√3 / 2) × 4(1 / √3)
= (2√3 / 2) × (4 / √3)
= √3 × (4 / √3)
= 4√3 / √3
= 4
Step 3: Add the results from Step 1 and Step 2.
Value = 3 + 4 = 7
The result is 7, which is an integer. The statement is shown to be true.
To show that the expression is an integer, we must evaluate it using the exact trigonometric values. These can be recalled or derived from a 30-60-90 right-angled triangle.
The exact values are:
- sin(30°) = 1/2
- cos(30°) = √3 / 2
- tan(30°) = 1 / √3
Now, substitute these into the expression:
`6 sin 30° + 2 cos 30° × 4 tan 30°`
`= 6(1/2) + 2(√3 / 2) × 4(1 / √3)`
We must follow the order of operations (BIDMAS), so we perform the multiplication first.
Let's evaluate the multiplication part: `2(√3 / 2) × 4(1 / √3)`
`= (2√3 / 2) × (4 / √3)`
The 2s in the first fraction cancel:
`= √3 × (4 / √3)`
Now, the √3 terms cancel:
`= 4`
Now substitute this back into the full expression:
`= 6(1/2) + 4`
`= 3 + 4`
`= 7`
The value of the expression is 7. Since 7 is an integer, we have shown what was required.
Common mistakes
✗ Not knowing or incorrectly recalling the exact trig values.
✗ Not following the order of operations (i.e., doing addition before multiplication).
✗ Making errors when multiplying or simplifying the fractions and surds. For example, √3 × 1/√3 = 3.
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