CALCU-The sine of an angle is given by the table b

Generated Graph

Solution by Steps

step 2

On the horizontal axis, each 1 cm represents

10^\circ

. On the vertical axis, each 1 cm represents 0.1 units of

sin(x)

step 3

Plot the points from the table on the graph using the given scale

step 4

Draw a smooth curve through the plotted points to represent the sine function from

0^\circ

to

90^\circ

step 5

To find

sin(55^\circ)

, locate

55^\circ

on the horizontal axis and find the corresponding point on the curve

step 6

Read the value of

sin(55^\circ)

from the vertical axis of the graph

[question 1] Answer

C

Key Concept

Graphical representation of sine function

Explanation

To find the sine of an angle using a graph, plot the known values and use the curve to estimate values between known points.

Solution by Steps

step 2

The calculator provides the exact value of

sin(55^\circ)

as well as the equivalent

cos(35^\circ)

step 3

The value of

sin(55^\circ)

is approximately 0.819152

[question 2] Answer

B

Key Concept

Sine of an angle

Explanation

The sine of an angle can be found using a calculator, which provides an exact or approximate value.

Solution by Steps

step 1

To calculate the gradient of line AB, we use the formula for the slope of a line between two points

(x_1, y_1)

and

(x_2, y_2)

, which is

\frac{y_2 - y_1}{x_2 - x_1}

step 2

Substituting the given points

A(2, 5)

and

B(-3, 4)

into the slope formula, we get

\frac{5 - 4}{2 - (-3)}

step 3

Simplifying the expression, we find the slope to be

\frac{1}{5}

1 Answer

The gradient of line AB is

\frac{1}{5}

or

0.2

.

step 1

To find the length of AB, we use the distance formula for two points

(x_1, y_1)

and

(x_2, y_2)

, which is

\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

step 2

Substituting the given points

A(2, 5)

and

B(-3, 4)

into the distance formula, we get

\sqrt{(2 - (-3))^2 + (5 - 4)^2}

step 3

Simplifying the expression, we find the distance to be

\sqrt{25 + 1}

step 4

The distance is

\sqrt{26}

, which is approximately

5.09902

2 Answer

The length of AB is

\sqrt{26}

or approximately

5.09902

.

Key Concept

Gradient and Distance between two points

Explanation

The gradient (or slope) of a line between two points is calculated using the change in y-coordinates divided by the change in x-coordinates. The distance between two points is found using the distance formula, which is the square root of the sum of the squares of the differences in x-coordinates and y-coordinates.

Find the equation of line A that passes through B (-2,0) and C (0,3)

Solution by Steps

step 1

To find the equation of a line, we need two points that the line passes through

step 2

We use the two given points B (-2,0) and C (0,3) to determine the slope (m) of the line

step 3

The slope m is calculated by the formula

m = \frac{y_2 - y_1}{x_2 - x_1}

where

(x_1, y_1)

and

(x_2, y_2)

are the coordinates of the two points

step 4

Substituting the given points into the slope formula gives

m = \frac{3 - 0}{0 - (-2)} = \frac{3}{2}

step 5

With the slope and one point, we can use the point-slope form of the equation of a line:

y - y_1 = m(x - x_1)

step 6

Substituting point B (-2,0) and the slope into the point-slope form gives

y - 0 = \frac{3}{2}(x - (-2))

step 7

Simplifying the equation gives

y = \frac{3}{2}x + 3

Answer

The equation of line A that passes through points B (-2,0) and C (0,3) is

y = \frac{3}{2}x + 3

.

Key Concept

Finding the equation of a line using two points

Explanation

The equation of a line can be found using the slope calculated from two points and then applying the point-slope form of the line equation.

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