Scottish Highers Maths Addition Formula: Triangle ABD is right-angled at

Triangle ABD is right-angled at B with angles BAC = p and BAD = q and lengths as shown in the diagram below - Scottish Highers Maths - Question 13 - 2016

Question 13

Triangle ABD is right-angled at B with angles BAC = p and BAD = q and lengths as shown in the diagram below. Show that the exact value of cos(q - p) is \( \frac{19}... show full transcript

Worked Solution & Example Answer:Triangle ABD is right-angled at B with angles BAC = p and BAD = q and lengths as shown in the diagram below - Scottish Highers Maths - Question 13 - 2016

Step 1

Calculate lengths AC and AD

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Answer

To find AC and AD, we use the given angle measures and the properties of right triangles. We know that:

For angle BAC (p):
- (AC = AB \cdot an(p))
- Given dimensions, calculate AC using the provided length of AB.
For angle BAD (q):
- (AD = AB \cdot an(q))
- Similarly, calculate AD using the length of AB.

Step 2

Select appropriate formula and express in terms of p and q

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Answer

Using the cosine difference identity, we can express (\cos(q - p)) as:

[ \cos(q - p) = \cos(q) \cos(p) + \sin(q) \sin(p) ]

Step 3

Calculate two of cos p, cos q, sin p, sin q

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Answer

Using the lengths calculated earlier:

(\cos(p) = \frac{AC}{AB} )
(\sin(p) = \frac{BC}{AB} )
(\cos(q) = \frac{AD}{AB} )
(\sin(q) = \frac{CD}{AB} )

Step 4

Calculate the other two and substitute into formula

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Answer

Continue from the above calculations:

Substitute the known values into the cosine difference identity:

[ \cos(q - p) = \left(\frac{AC}{AB}\right) \left(\frac{AD}{AB}\right) + \left(\frac{BC}{AB}\right) \left(\frac{CD}{AB}\right) ]

Simplifying this will yield the final expression in terms of the lengths and angles.

Step 5

Arrange into required form

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Answer

After simplifying, we achieve:

[ \cos(q - p) = \frac{19}{85} ]

This shows that the exact value of ( \cos(q - p) ) is successfully derived.

Triangle ABD is right-angled at B with angles BAC = p and BAD = q and lengths as shown in the diagram below - Scottish Highers Maths - Question 13 - 2016

Worked Solution & Example Answer:Triangle ABD is right-angled at B with angles BAC = p and BAD = q and lengths as shown in the diagram below - Scottish Highers Maths - Question 13 - 2016

Calculate lengths AC and AD

Select appropriate formula and express in terms of p and q

Calculate two of cos p, cos q, sin p, sin q

Calculate the other two and substitute into formula

Arrange into required form

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