Show that
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Show that
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Use the identity
coscos cos sin sin
to show that
cos cos2 sin2
Show by counter-example that
cos cos cos
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Given that is small and measured in radians, use an appropriate approximation to show that
sin cos
Use the result in part (a) to find an approximation to sin cos.
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Prove the identity
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Show that
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Show that
sin sin sin
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Prove the identity
cosec cot cosec2
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Find the value of
Explain why the answer to part (a) (ii) is not .
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Use the identity
sincos sin sin cos
to show that
sin
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Given the identity
sin2 cos2
prove the following identities:
sec2 tan2
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cos cos4 cos2
sin sin4 sin2
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Given that is small, and that terms involving or higher powers of can be ignored, use an appropriate approximation to show that
cos cos2
Show that the result in part (a) gives a percentage error of 0.583%, to 3 significant figures, when used to approximate
coscos2
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Prove the identity
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Show that
sin
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Use the quotient rule to show that
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Show that
sin sin sin sin3
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Prove the identity
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Show that
sin
Show that
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Show that
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Given the identity
coscos cos sin sin
prove the following identities:
cos cos2 sin2
cos sin2
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sin sin sin3
cos cos cos3
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Given that is small, and that terms involving or higher powers of can be ignored, show that
Determine the percentage error when the result in part (a) is used to approximate
giving your answer correct to 3 significant figures.
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Show that
cos cos sin4 sin2
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Prove that
cot2 tan2 cot cosec .
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Prove the identity
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Prove the identity
cosec
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Show that
tan
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Given that , where is a constant, show that .
Hence show that the value of is constant and independent of .
Find the value of this constant.
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Show that
where
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Consider the three triangles, all of height 1, as shown below.
By applying the area of a triangle formula to each one, prove that,
sinsin cos sin cos
Briefly explain why this only proves the result for and being acute angles.
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Prove the identity
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Use the small angle approximations for sine and cosine to confirm the following two limit results:
Be sure to explain why use of the small angle approximations is justified here.
Hence prove from first principles that
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Prove the identity
cot cosec3 sec4 cosec4
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Show that
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Show that
sin sin cos2 sin3
Hence, or otherwise, show that
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Show that
cos2 sin2 sin
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Show that
tansec tan
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On the axes below sketch the graphs of
and
With the help of your sketch, determine the exact solution(s) to the equation
What can you say about the solution(s) to the equation
Justify your answer.
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Show that
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