Determine and interpret the six trigonometric ratios for angles in standard position

Alberta Grade 12 Mathematics

Trigonometry

Use a point on an angle’s terminal side to determine and interpret each ratio.

Trigonometric ratios connect an angle to the coordinates of a point on its terminal side. They can describe direction as well as a ratio of lengths. In this lesson, you will use one set of coordinate rules for angles in any quadrant. You will also learn why some ratios are undefined for certain angles.

What you will learn

Identify an angle in standard position and locate a point on its terminal side.
Use the coordinates of a point to determine sine, cosine, and tangent.
Determine cosecant, secant, and cotangent as reciprocal ratios.
Interpret the signs and identify when a trigonometric ratio is undefined.

Review: coordinates, distance, and angle position

A point is written as an ordered pair, such as

(x,y)

. The first coordinate,

x

, tells how far the point is horizontally from the origin. The second coordinate,

y

, tells how far it is vertically from the origin. The origin is

(0,0)

.

An angle is in standard position when its vertex is at the origin and its initial side lies along the positive

x

-axis. Its terminal side is the ray reached after turning through the angle. A positive angle turns counterclockwise; a negative angle turns clockwise.

A point

(x,y)

on the terminal side helps describe the angle. The distance from the origin to the point is called

r

. The distance formula gives

r=\sqrt{x^2+y^2}

. Since

r

is a distance, it is positive for any point other than the origin.

r=\sqrt{x^2+y^2}

The point used for the ratios must lie on the angle’s terminal side, but it does not have to be a particular distance from the origin.
The origin itself cannot be used because its distance $r$ is zero.

The six ratios and their signs

For a point

(x,y)

on the terminal side of an angle, sine, cosine, and tangent are defined by coordinates. The other three ratios are their reciprocals. A reciprocal is found by reversing the numerator and denominator of a fraction.

The coordinate rules work in all four quadrants. The signs come from the signs of

x

and

y

. For example, a point in quadrant II has negative

x

and positive

y

. Its sine is positive, its cosine is negative, and its tangent is negative.

The reciprocal ratios have the same signs as the ratios they reciprocate. Cosecant has the sign of sine, secant has the sign of cosine, and cotangent has the sign of tangent. This makes the point’s quadrant a useful check on your answers.

\sin\theta=\frac{y}{r},\quad \cos\theta=\frac{x}{r},\quad \tan\theta=\frac{y}{x},\quad \csc\theta=\frac{r}{y},\quad \sec\theta=\frac{r}{x},\quad \cot\theta=\frac{x}{y}

Sine uses $y$ and $r$ ; cosine uses $x$ and $r$ ; tangent uses $y$ and $x$ .
Cosecant, secant, and cotangent are reciprocals of sine, cosine, and tangent, respectively.
A ratio is undefined if its denominator is zero.

Restrictions and a strategy

The coordinate rules also show when a ratio is undefined. If

x=0

, then tangent and secant are undefined because their denominators are zero. If

y=0

, then cosecant and cotangent are undefined. Sine and cosine are defined for every angle because

r

is positive for a point on a terminal side other than the origin.

Use this order when solving a problem: find or select a point on the terminal side, calculate

r

, determine the six ratios, and check their signs against the quadrant. If a problem gives an angle instead of a point, use its direction to choose a convenient point on the terminal side. Any non-origin point on that ray gives the same ratios.

Exact values, such as fractions or square roots, show the value without rounding. A decimal marked with an approximation symbol is an estimate. When a point and its coordinates are given, keep exact forms unless the question asks for a decimal.

x=0:\ \tan\theta,\sec\theta\text{ undefined};\qquad y=0:\ \csc\theta,\cot\theta\text{ undefined}

Check denominators before stating a ratio.
Use the quadrant to check signs, not to change the coordinate definitions.
Do not confuse an exact ratio with a rounded decimal approximation.

Practice the routine

Before looking at the worked solutions, try to state the six ratios for each point or angle. For a coordinate point, first calculate the distance

r

. For a given angle, sketch its direction mentally or on paper and choose a point on its terminal side.

After calculating, check three things: whether the signs match the quadrant, whether any denominator is zero, and whether each reciprocal pair multiplies to

1

when both values are defined.

Keep the point’s coordinates in the order $(x,y)$ .
A reciprocal check can help catch a fraction that was inverted incorrectly.

Signs of the six ratios by quadrant

Quadrant	sin and csc	cos and sec	tan and cot
I	positive	positive	positive
II	positive	negative	negative
III	negative	negative	positive
IV	negative	positive	negative

Worked example

A point in quadrant II

A point

P(-8,6)

lies on the terminal side of an angle

\theta

in standard position. Determine all six trigonometric ratios of

\theta

.

Angle direction in quadrant II

The terminal side is in quadrant II; the displayed circle indicates direction, not the coordinates of point $P$.

Find the distance from the origin
The point is in quadrant II because its $x$ -coordinate is negative and its $y$ -coordinate is positive. Calculate its distance $r$ from the origin using both coordinates.
$r=\sqrt{(-8)^2+6^2}=10$
Use the coordinate definitions
Substitute $x=-8$ , $y=6$ , and $r=10$ into the six definitions. The signs fit quadrant II: sine is positive, cosine is negative, and tangent is negative.
$\sin\theta=\frac{3}{5},\quad \cos\theta=-\frac{4}{5},\quad \tan\theta=-\frac{3}{4},\quad \csc\theta=\frac{5}{3},\quad \sec\theta=-\frac{5}{4},\quad \cot\theta=-\frac{4}{3}$

Answer: The six ratios are

\sin\theta=\frac{3}{5}

,

\cos\theta=-\frac{4}{5}

,

\tan\theta=-\frac{3}{4}

,

\csc\theta=\frac{5}{3}

,

\sec\theta=-\frac{5}{4}

, and

\cot\theta=-\frac{4}{3}

.

Check: The products of the reciprocal pairs are

\sin\theta\csc\theta=1

,

\cos\theta\sec\theta=1

, and

\tan\theta\cot\theta=1

.

Worked example

Use a ratio and quadrant to find all six

An angle

\alpha

is in quadrant IV, and

\tan\alpha=-\frac{5}{12}

. Determine the other five ratios and verify the tangent value.

Angle direction in quadrant IV

The terminal side is in quadrant IV; the displayed circle indicates direction, not a scale drawing of the chosen point.

Choose coordinates consistent with the information
Tangent is $y/x$ . In quadrant IV, $x$ is positive and $y$ is negative. Choose $x=12$ and $y=-5$ , which gives the stated tangent.
$\frac{y}{x}=-\frac{5}{12}$
Calculate the distance and ratios
The distance is $13$ . Using the coordinates and distance gives the sine and cosine values; reversing each fraction gives its reciprocal. The signs agree with quadrant IV.
$r=\sqrt{12^2+(-5)^2}=13,\quad \sin\alpha=-\frac{5}{13},\quad \cos\alpha=\frac{12}{13},\quad \csc\alpha=-\frac{13}{5},\quad \sec\alpha=\frac{13}{12},\quad \cot\alpha=-\frac{12}{5}$

Answer: The other five ratios are

\sin\alpha=-\frac{5}{13}

,

\cos\alpha=\frac{12}{13}

,

\csc\alpha=-\frac{13}{5}

,

\sec\alpha=\frac{13}{12}

, and

\cot\alpha=-\frac{12}{5}

. Tangent is

-\frac{5}{12}

as given.

Check: In quadrant IV, sine and tangent are negative, while cosine is positive. The results have those signs.

Worked example

An angle with a diagonal terminal side

Determine the six ratios for

\theta=225^\circ

.

Angle of 225 degrees

The terminal side points into quadrant III. The point used in the calculation lies in the same direction.

Locate the terminal side
The angle turns counterclockwise past $180^\circ$ , so its terminal side is in quadrant III. It lies equally far below and to the left of the origin. Choose the point $(-1,-1)$ on that side.
$x=-1,\quad y=-1$
Calculate all six ratios
The point’s distance is $\sqrt{2}$ . Substitute into the definitions and simplify. Both sine and cosine are negative in quadrant III, so tangent is positive.
$r=\sqrt{2},\quad \sin\theta=-\frac{\sqrt{2}}{2},\quad \cos\theta=-\frac{\sqrt{2}}{2},\quad \tan\theta=1,\quad \csc\theta=-\sqrt{2},\quad \sec\theta=-\sqrt{2},\quad \cot\theta=1$

Answer: The six ratios are

\sin\theta=-\frac{\sqrt{2}}{2}

,

\cos\theta=-\frac{\sqrt{2}}{2}

,

\tan\theta=1

,

\csc\theta=-\sqrt{2}

,

\sec\theta=-\sqrt{2}

, and

\cot\theta=1

.

Check: The point is in quadrant III, where sine and cosine are negative and tangent is positive. The signs match.

Common mistakes and how to avoid them

Using

x

as the numerator for sine or

y

as the numerator for cosine.

Correction: Sine uses

y/r

and cosine uses

x/r

. Check the coordinate definitions before substituting.

Making all six ratios positive because distances are positive.

Correction:

r

is positive, but

x

and

y

can be negative. The coordinate signs determine the ratio signs.

Giving a value for tangent when

x=0

.

Correction: Tangent has denominator

x

. Division by zero is undefined, so tangent is undefined when

x=0

.

Finding a reciprocal without reversing the fraction.

Correction: For a nonzero ratio, reverse its numerator and denominator to get its reciprocal. For example, if sine is

3/5

, cosecant is

5/3

.

Lesson summary

For a point $(x,y)$ on a terminal side, calculate $r=\sqrt{x^2+y^2}$ .
Use $\sin\theta=y/r$ , $\cos\theta=x/r$ , and $\tan\theta=y/x$ .
Use $\csc\theta=r/y$ , $\sec\theta=r/x$ , and $\cot\theta=x/y$ .
Use the quadrant to check signs, and check denominators to identify undefined ratios.

Check your understanding

Question 1

A point

(3,-4)

lies on the terminal side of

\beta

. What is

\cos\beta

?

$-\frac{3}{5}$
$\frac{3}{5}$
$-\frac{4}{5}$
$\frac{4}{5}$

Show answer and explanation

\frac{3}{5}

The distance is

5

, and cosine is

x/r

. Therefore,

\cos\beta=3/5

.

Question 2

Which ratio is undefined when a terminal side lies on the positive

y

-axis?

Sine
Cosine
Cosecant
Secant

Show answer and explanation

Secant

On the positive

y

-axis,

x=0

. Secant is

r/x

, so it is undefined. Cosine is zero, but it is defined.

Question 3

An angle is in quadrant III. Which sign pattern is correct for sine, cosine, and tangent?

Positive, positive, positive
Positive, negative, negative
Negative, negative, positive
Negative, positive, negative

Show answer and explanation

Negative, negative, positive

In quadrant III, both

x

and

y

are negative. Thus

y/r

and

x/r

are negative, while

y/x

is positive.

Key terms

Angle in standard position: An angle whose vertex is at the origin and whose initial side lies along the positive $x$ -axis.
Terminal side: The ray showing the direction reached after turning through an angle from its initial side.
Quadrant: One of the four regions formed by the $x$ - and $y$ -axes.
Reciprocal: A value formed by reversing the numerator and denominator of a nonzero fraction.

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Published by DoAssignment. This reviewed lesson follows Alberta Grade 12 Mathematics (Math 30-1), expectation TR3. It is a study resource, not an official curriculum publication.

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TR3 · Determine and interpret the six trigonometric ratios for angles in standard position

Alberta Grade 12 Mathematics

Use a point on an angle’s terminal side to determine and interpret each ratio.

What you will learn

Review: coordinates, distance, and angle position

The six ratios and their signs

Restrictions and a strategy

Practice the routine

Signs of the six ratios by quadrant

A point in quadrant II

Use a ratio and quadrant to find all six

An angle with a diagonal terminal side

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Math 30-1

About this lesson and its review