An exponential graph y=ax (with a>1) passes through (0,1) and climbs ever more steeply. y=sinx and y=cosx repeat every 360∘ between −1 and 1;y=tanx repeats every 180∘.
Plot key points: (0,1) for exponentials; 0∘,90∘,180∘,270∘,360∘ for trig.
y=tanx has asymptotes at 90∘ and 270∘.
Use the symmetry of the graph to find other solutions.
Watch outsin and cos never go above 1 or below −1.
See it worked
Worked exampleCalculator allowed·2 marks
A model gives the mass of a culture at times 0, 1, 2, 3 and 4 hours as 80, 120, 180, 270 and 405 grams respectively. The plotted points are joined by straight segments. Use this graph model to estimate when the mass reaches 300 grams. Give your answer to 1 decimal place.
3+(300−270)/(405−270)
300 grams lies between the points (3, 270) and (4, 405).
The increase of 30 grams is 30/(405 − 270) = 2/9 of this one-hour segment.
The estimate is 3 + 2/9 = 3.222… hours, or 3.2 hours to 1 decimal place.
Answer3.2 hours
Your turn
Question 1Calculator allowed·2 marks
Find all values of x between 0∘ and 360∘, inclusive, for which sin x = 1/2.
Question 2Non-calculator·5 marks
y=2sin(x−60) for 60∘≤x≤420∘.
(a) Find all x for which y = 1.
(b) Find the maximum value of y.
Question 3Non-calculator·5 marks
sin40∘=0.643, correct to 3 decimal places.
(a) Find another angle between 0∘ and 360∘ whose sine is 0.643.
(b) Find the angle between 180∘ and 270∘ whose sine is −0.643.
(c) Find the angle x between 0∘ and 90∘ with cosx=0.643.
In past papers
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