Questions With Solutions

How to use the unit circle to find properties and trigonometric identities of the sine and
cosine functions? Grade 11 trigonometry questions are presented along with detailed solutions and explanations.

A circle has an infinite number of symmetries with respects to lines through the center and a symmetry with respect to its center. We are interested here on the symmetries with respect to its center, the x-axis, the y-axis an the line y = x. It will be shown how the use of these symmetries allows us to write several identities in trigonometry.

Four angles (?, ? - ?, ? + ? and 2? - ?) are shown below in a unit circle. To each angle corresponds a point (

.

The four angles have the same reference angle equal to ?. Because of the symmetry of the circle, the four points form a rectangle ABCD as shown above. Points A and B are reflection of each other of the y-axis. Points A and C are reflection of each other on the origin of the system of axis. Points A and D are reflection of each other on the x-axis. Given the coordinates

A: (a , b) , B: (- a , b), C: (- a , - b) and D: (a , - b)

We now express the coordinates of each point in terms of the sine and cosine of the corresponding angle as follows.

A: (a , b) = (cos ? , sin ?)

B: (- a , b) = (cos(? - ?) , sin(? - ?))

C: (- a , - b) = (cos(? + ?) , sin(? + ?))

D: (a , - b) = (cos(2? - ?) , sin(2? - ?))

cos(? - ?) = - cos ?

sin(? - ?) = sin ?

Comparing the x and y-coordinates of points A and C, we can write

cos(? + ?) = - cos ?

sin(? + ?) = - sin ?

Comparing the x and y-coordinates of points A and D, we can write

cos(2? - ?) = cos ?

sin(2? - ?) = - sin ?

.

Points A and D are reflection of each other on the x-axis. Given the coordinates

D: (a , - b)

We now express the coordinates of points A and D in terms of the sine and cosine of the corresponding angle as follows.

A: (a , b) = (cos ? , sin ?)

D: (a , - b) = (cos(- ?) , sin(- ?))

Examples of Identities that may be Deduced

cos(- ?) = cos ?

sin( - ?) = - sin ?

.

Points A and B are reflection of each other on the line y = x. Given the coordinates

B: (b , a)

We now express the coordinates of points A and B in terms of the sine and cosine of the corresponding angle as follows.

A: (a , b) = (cos ? , sin ?)

B: (b , a) = (cos(?/2 - ?) , sin(?/2 - ?))

Examples of Identities that may be Deduced

cos(?/2 - ?) = sin ?

sin(?/2 - ?) = cos ?

Use the following general identities

1) cos (A + B) = cos A cos B - sin A sin B

2) cos (A - B) = cos A cos B + sin A sin B

3) sin(A + B) = sin A cos B + cos A sin B

4) sin(A - B) = sin A cos B - cos A sin B

to verify the identities listed below.

- cos(? - ?) = - cos ?
- sin(? - ?) = sin ?
- cos(? + ?) = - cos ?
- sin(? + ?) = - sin ?
- cos(2? - ?) = cos ?
- sin(2? - ?) = - sin ?
- cos(- ?) = cos ?
- sin( - ?) = - sin ?
- cos(?/2 - ?) = sin ?
- sin(?/2 - ?) = cos ?

- Use the general identity cos (A - B) = cos A cos B + sin A sin B to expand cos(? - ?) as follows:

cos(? - ?) = cos ? cos ? + sin ? sin ?

Use cos ? = - 1 and sin ? = 0 to simplify the above to

= - cos ?)

- Use the general identity sin(A - B) = sin A cos B - cos A sin B to expand sin(? - ?) as follows:

sin(? - ?) = sin ? cos ? - cos ? sin ?

Use sin ? = 0 and cos ? = - 1 to simplify the above to

= sin ?

- Use the general identity cos (A + B) = cos A cos B - sin A sin B to expand cos(? + ?) as follows:

cos(? + ?) = cos ? cos ? - sin ? sin ?

Use cos ? = - 1 and sin ? = 0 to simplify the above to

= - cos ?)

- Use the general identity sin(A + B) = sin A cos B + cos A sin B to expand sin(? + ?) as follows:

sin(? + ?) = sin ? cos ? + cos ? sin ?

Use sin ? = 0 and cos ? = - 1 to simplify the above to

= - sin ?

- Use the general identity cos (A - B) = cos A cos B + sin A sin B to expand cos(2? - ?) as follows:

cos(2? - ?) = cos 2? cos ? + sin 2? sin ?

Use cos 2? = 1 and sin 2? = 0 to simplify the above to

= cos ?)

- Use the general identity sin(A - B) = sin A cos B - cos A sin B to expand sin(2? - ?) as follows:

sin(2? - ?) = sin 2? cos ? - cos 2? sin ?

Use sin 2? = 0 and cos 2? = 1 to simplify the above to

= - sin ?

- We first write the left side of the identity to verify cos(- ?) = cos ? as follows:

cos(- ?) = cos(0 - ?)

We then use the general identity cos (A - B) = cos A cos B + sin A sin B to expand cos(0 - ?) as follows:

cos(- ?) = cos(0 - ?) = cos 0 cos ? + sin 0 sin ?

Use cos 0 = 1 and sin 0 = 0 to simplify the above to

= cos ?)

- We first write the left side of the given identity to verify sin( - ?) = - sin ? as follows:

sin( - ?) = sin (0 - ?)

We then use the general identity sin(A - B) = sin A cos B - cos A sin B to expand sin(0 - ?) as follows:

sin( - ?) = sin(0 - ?) = sin 0 cos ? - cos 0 sin ?

Use sin 0 = 0 and cos 0 = 1 to simplify the above to

= - sin ?

- Use the general identity cos (A - B) = cos A cos B + sin A sin B to expand cos(?/2 - ?) as follows:

cos(?/2 - ?) = cos ?/2 cos ? + sin ?/2 sin ?

Use cos ?/2 = 0 and sin ?/2 = 1 to simplify the above to

= sin ?)

- Use the general identity sin(A - B) = sin A cos B - cos A sin B to expand sin(?/2 - ?) as follows:

sin(?/2- ?) = sin ?/2 cos ? - cos ?/2 sin ?

Use sin ?/2 = 1 and cos ?/2 = 0 to simplify the above to

= cos ?

Verify Trigonometric Identities

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