Grade 12 trigonometry problems and questions on how to find the period of trigonometric functions given its graph or formula, are presented along with detailed solutions.

In the problems below, we will use the formula for the period P of trigonometric functions of the form y = a sin(bx + c) + d or y = a cos(bx + c) + d and which is given by

Interactive tutorials on Period of trigonometric functions may first be used to understand this concept.

Locate two zeros that delimit a whole cycle or an integer number of cycles. In this example, we can see that from the zero at x = 0 to the zero at x = 1, there are two cycles. Hence the period P is equal to:

P = (1 - 0) / 2 = 1 / 2

We now calculate b by equating the value of the period found using the graph to the above formula and solve for b.

1 / 2 = 2? / b

b = 4 ?

There is one cycle from the zero at x = -?/4 to the zero at x = ?/4. Hence the period P is equal to:

P = ?/4 - (-?/4) = ?/2

We now equate the value of the period found using the graph to the above formula and solve for b.

?/2 = 2? / b

b = 4

There are two zeros that delimit half a cycle. We first find these zeros.

Zero on the left: (-? / 4 - ? / 8 ) / 2 = - 3? / 16 (assuming it is in the middle of x = -? / 4 and -? / 8)

Zero on the right: (0 + ? / 8 ) / 2 = ? / 16 (assuming it is in the middle of x = 0 and ? / 8)

Hence half a period is equal to:

(? / 16 - (- 3? / 16)) = ? / 4

and a period P is equal to:

P = 2 × ? / 4 = ? / 2

We now equate the value of the period found using the graph to the above formula and solve for b.

?/2 = 2? / b

b = 4

The distance along the x axis between points A and B is equal to half a period and is given by

7? / 6 - 3? / 6 = 2 ? / 3

The period P of the function is given by

P = 2× 2 ? / 3 = 4 ? / 3

b is found by solving

2 ? / b = 4 ? / 3

b = 3 / 2

The is one whole cycle between points A and B. Hence period P is given by

P = 0.1 - (-0.3) = 0.4

b is found by solving

2 ? / b = 0.4

b = 5?

1) y = sin(x)cos(x) - 3

2) y = 2 + 5 cos

3) y = cos(x) + sin(x)

1) Use the identity sin(2x) = 2 sin(x)cos(x) to rewrite the given function as follows:

y = (1 / 2) sin(2x) - 3

Use the formula P = 2? / b to find the period as

P = 2? / 2 = ?

2) Use the identity cos

y = 2 + 5 cos

Use the formula P = 2? / b to find the period as

P = 2? / 2 = ?

3) Rewrite the given function as follows:

y = cos(x) + sin(x) = (2 / √2)(√2 / 2 cos(x) + √2 / 2 sin(x))

Use the identity:

sin(? / 4 + x) = sin(? / 4) cos(x) + cos(? / 4) sin(x) = √2 / 2 cos(x) + √2 / 2 sin(x)

to rewrite the given function as:

y = cos(x) + sin(x) = (2 / √2) sin(x + ? / 4)

Use the formula P = 2? / b to find the period as

P = 2? / 1 = 2 ?

If p is the period of function f, then

f(x + p) = f(x) for all x in the domain of f.

Let x = k X , where k is a constant.

f(k X + p) = f(k X)

Rewrite the above as

f(k(X + p / k)) = f (k X)

Let h(x) = f(k x). The above may be written as

h(X + p / k) = h(X)

Which indicates that h(x) = f(k x) is periodic and has a period equal to p / k.

Properties of The Six Trigonometric Functions

Interactive tutorials on Period of trigonometric functions.

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