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Access custom-built, exam-style problems for trig equations & identities. Each problem has a full solution and mark-scheme, as well as AI grading and support.
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The temperature, T degrees Fahrenheit, t hours after sunrise in Boston can be modeled by a function of the form
The following diagram shows the curve of T on a day in the middle of each of the 4 seasons.
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Determine which curve corresponds to which season to complete the table below.
Explain why a must be a negative number.
Find the value of b.
It is given that the equation of curve C is T(t)=−10cos(bt°)+45, where b has the value determined in (b.ii).
Write down the average temperature throughout the day.
Find the second time at which the temperature is 40°F.
It is given that the temperature on the winter day T(t)=−9cos(bt°)+29. The formula for converting Fahrenheit to Celsius is C=95(F−32).
Find an expression for the temperature C(t), in Celsius, throughout the winter day.
Hence determine the length of time during the day where the temperature is below freezing (0°C).
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Consider the function g defined by g(x)=2−4sin2x. The following graph shows the curve y=g(x) for 0<x<23π.
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The graph of y=g(x) intercepts the x-axis at a, b, c and d, as shown.
Find the values of a,b,c and d.
In the diagram above, the region enclosed by the curve of y=g(x) and the x-axis between x=b and x=c is shaded.
Show that the area of the shaded region is 34π+2√3.
The diagram below shows a circle of radius r circumscribing a shape whose area is 34π+2√3.
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Find the value of r.
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Let w1, w2 and w3 be the roots of the equation z3=1, where 0<arg(w1)<arg(w2)<arg(w3)<2π.
Find in the form a+bi
(i) w1
(ii) w2
(iii) w3
The complex numbers w1,w2 and w3 are represented by the points P1,P2 and P3, respectively, on an Argand diagram. The following diagram shows triangle P1P2P3.
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Find the area of the triangle.
A circle, centered at the origin, is now inscribed within the triangle, as shown in the diagram above.
Show that the circle has unit diameter.
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Show that log3√4−3sin2x=log9(4−3sin2x).
It is given that log9(4cosx)=log3√4−3sin2x and 0<x<2π
Find the value of tanx.
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