In the diagram are the graphs of $f(x) = ext{sin} 2x$ and $h(x) = ext{cos}(x - 45^{ ext{o}})$ for the interval $x ext{ }
i ext{ }[-180^{ ext{o}} ; 180^{ ext{o}}]$ - NSC Mathematics - Question 6 - 2017 - Paper 2
Question 6
In the diagram are the graphs of $f(x) = ext{sin} 2x$ and $h(x) = ext{cos}(x - 45^{ ext{o}})$ for the interval $x ext{ }
i ext{ }[-180^{ ext{o}} ; 180^{ ext{o}... show full transcript
Worked Solution & Example Answer:In the diagram are the graphs of $f(x) = ext{sin} 2x$ and $h(x) = ext{cos}(x - 45^{ ext{o}})$ for the interval $x ext{ }
i ext{ }[-180^{ ext{o}} ; 180^{ ext{o}}]$ - NSC Mathematics - Question 6 - 2017 - Paper 2
Step 1
Write down the period of $f$
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Answer
The period of the function f(x)=extsin2x is given by the formula:
ext{Period} = rac{360^{ ext{o}}}{k}
where k is the coefficient of x in the sine function. For f(x), k=2, so:
ext{Period} = rac{360^{ ext{o}}}{2} = 180^{ ext{o}}.
Step 2
Determine the x-coordinate of B.
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Answer
From the provided graph, point B is located at −75exto. This can be inferred from the intersections of the two graphs, where we look for the x value within the given range.
Step 3
Use the graphs to solve $2 ext{sin} x ext{cos} x ext{ } rac{1}{ ext{√}2}( ext{cos} x + ext{sin} x)$ for the interval $x ext{ } ext{ }
i ext{ }[-180^{ ext{o}} ; 180^{ ext{o}}]$. Show all working.
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Answer
To solve the equation, we start with:
2 ext{sin} x ext{cos} x = rac{1}{ ext{√}2}( ext{cos} x + ext{sin} x).
We can rewrite the left side using the double angle identity:
ext{sin}(2x) = rac{1}{ ext{√}2}( ext{cos} x + ext{sin} x).
Next, we rearrange and analyze the function over the interval [−180exto;180exto]. To solve for x, we can break it down further:
Identify critical angles where both the left and right sides equate.
Substitute known values into extsin(2x) to find potential solutions.
Use the properties of trig functions to determine valid values of x, ensuring they lie within the specified interval.
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