Calculus of Trigonometric Functions

NZ Level 8 (NZC) Level 3 (NCEA) [In development]

Basic derivatives of sine and cosine

Consider the graph of $y=\sin x$`y`=`s``i``n``x`.

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a

Which of the following best describes the graph of $y=\sin x$`y`=`s``i``n``x`?

It is constantly increasing.

A

It is constantly decreasing.

B

The graph increases and decreases periodically.

C

It is constantly increasing.

A

It is constantly decreasing.

B

The graph increases and decreases periodically.

C

b

Which of the following best describes the nature of the gradient of the curve?

Select all the correct options.

Between points where the gradient is $0$0, the gradient is always negative.

A

The gradient of the curve is $0$0 once every $2\pi$2π radians.

B

Between points where the gradient is $0$0, the gradient is always positive.

C

The gradient of the curve is $0$0 once every $\pi$π radians.

D

Between points where the gradient is $0$0, the gradient is positive and negative alternately.

E

Between points where the gradient is $0$0, the gradient is always negative.

A

The gradient of the curve is $0$0 once every $2\pi$2π radians.

B

Between points where the gradient is $0$0, the gradient is always positive.

C

The gradient of the curve is $0$0 once every $\pi$π radians.

D

Between points where the gradient is $0$0, the gradient is positive and negative alternately.

E

c

Select all the intervals in which the gradient of $y=\sin x$`y`=`s``i``n``x` is positive.

$\frac{\pi}{2}`x`≤π

A

$\pi\le x<\frac{3\pi}{2}$π≤`x`<3π2

B

$\frac{3\pi}{2}`x`≤2π

C

$0\le x<\frac{\pi}{2}$0≤`x`<π2

D

$\frac{\pi}{2}`x`≤π

A

$\pi\le x<\frac{3\pi}{2}$π≤`x`<3π2

B

$\frac{3\pi}{2}`x`≤2π

C

$0\le x<\frac{\pi}{2}$0≤`x`<π2

D

d

Select all the intervals in which the gradient of $y=\sin x$`y`=`s``i``n``x` is negative.

$\frac{3\pi}{2}`x`≤2π

A

$0\le x<\frac{\pi}{2}$0≤`x`<π2

B

$\frac{\pi}{2}`x`≤π

C

$\pi\le x<\frac{3\pi}{2}$π≤`x`<3π2

D

$\frac{3\pi}{2}`x`≤2π

A

$0\le x<\frac{\pi}{2}$0≤`x`<π2

B

$\frac{\pi}{2}`x`≤π

C

$\pi\le x<\frac{3\pi}{2}$π≤`x`<3π2

D

e

The gradient function $y'$`y`′ is to be graphed on the axes below. The plotted points correspond to where the gradient of $y=\sin x$`y`=`s``i``n``x` is $0$0.

Given that the gradient at $0$0 is $1$1, graph the gradient function $y'$`y`′.

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f

Which of the following is the equation of the gradient function $y'$`y`′ graphed in the previous part?

$y'=-\cos x$`y`′=−`c``o``s``x`

A

$y'=-\sin x$`y`′=−`s``i``n``x`

B

$y'=\sin x$`y`′=`s``i``n``x`

C

$y'=\cos x$`y`′=`c``o``s``x`

D

$y'=-\cos x$`y`′=−`c``o``s``x`

A

$y'=-\sin x$`y`′=−`s``i``n``x`

B

$y'=\sin x$`y`′=`s``i``n``x`

C

$y'=\cos x$`y`′=`c``o``s``x`

D

Easy

Approx 6 minutes

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