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Trig Ratios in all Four Quadrants (2)

In this worksheet, students match trigonometric functions by looking at the quadrants in which the angles lie.

'Trig Ratios in all Four Quadrants (2)' worksheet

Key stage:  KS 4

Curriculum topic:  Ratio, Proportion and Rates of Change

Curriculum subtopic:  Compare Lengths, Areas and Volumes Using Ratio Notation/Scale Factors

Difficulty level:  

down

Worksheet Overview

QUESTION 1 of 10

In this worksheet, you will work with trigonometric functions of any angle using the ASTC circle to determine in which quadrants the functions are positive, before finding "equivalent" trig ratios of angles less than 90º.

Remember that angles are measured anticlockwise from the 0º line.

 

The following diagram shows the angle 200º on the ASTC circle.

 

 

Notice that the 200º line makes an angle of 20º with the horizontal.

We always look at the horizontal.

 

This shows that:

tan 200º = tan 20º as tan is positive in this 3rd quadrant.

sin 200º = - sin 20º as sin is negative in this 3rd quadrant.

cos 200º = -cos 20º as cos is negative in this 3rd quadrant.

 

Example

Change the following to equivalent trigonometric functions of angles less than 90º:

sin 165º, cos 165º and tan 165º

 

Answer

Draw 165º on the ASTC circle and then put in the angle that it makes with the horizontal.

 

The only trig ratio that is positive in this quadrant is sin 165º.

 

sin 165º = sin 15º as sin is positive in this 2nd quadrant.

cos 165º = - cos 15º as cos is negative in this 2nd quadrant.

tan 165º = - tan 15º as tan is negative in this 2nd quadrant.

Change the following to an equivalent trigonometric function of an angle less than 90º:

 

sin 170º

sin 10°

-sin 10°

sin 80°

-sin 80°

Change the following to an equivalent trigonometric function of an angle less than 90º:

 

cos 170º

cos 10°

-cos 10°

cos 80°

-cos 80°

Change the following to an equivalent trigonometric function of an angle less than 90º:

 

tan 170º

tan 10°

-tan 10°

tan 80°

-tan 80°

Change the following to an equivalent trigonometric function of an angle less than 90º:

 

cos 230º

cos 40°

-cos 40°

cos 50°

-cos 50°

Change the following to an equivalent trigonometric function of an angle less than 90º:

 

tan 230º

tan 40°

-tan 40°

tan 50°

-tan 50°

Change the following to an equivalent trigonometric function of an angle less than 90º:

 

sin 280º

sin 10°

-sin 10°

sin 80°

-sin 80°

Change the following to an equivalent trigonometric function of an angle less than 90º:

 

cos 280º

cos 10°

-cos 10°

cos 80°

-cos 80°

Change the following to an equivalent trigonometric function of an angle less than 90º:

 

cos 132º

cos 48°

-cos 48°

cos 42°

-cos 42°

Change the following to an equivalent trigonometric function of an angle less than 90º:

 

sin 365º

sin 5°

-sin 5°

sin 85°

-sin 85°

Change the following to an equivalent trigonometric function of an angle less than 90º:

 

tan 580º

tan 40°

-tan 40°

tan 50°

-tan 50°

  • Question 1

Change the following to an equivalent trigonometric function of an angle less than 90º:

 

sin 170º

CORRECT ANSWER
sin 10°
EDDIE SAYS
The 170° line makes an angle of 10° with the horizontal.
170° is in the 2nd quadrant where sin is positive.
  • Question 2

Change the following to an equivalent trigonometric function of an angle less than 90º:

 

cos 170º

CORRECT ANSWER
-cos 10°
EDDIE SAYS
The 170° line makes an angle of 10° with the horizontal.
170° is in the 2nd quadrant where cos is negative.
  • Question 3

Change the following to an equivalent trigonometric function of an angle less than 90º:

 

tan 170º

CORRECT ANSWER
-tan 10°
EDDIE SAYS
The 170° line makes an angle of 10° with the horizontal.
170° is in the 2nd quadrant where tan is negative.
  • Question 4

Change the following to an equivalent trigonometric function of an angle less than 90º:

 

cos 230º

CORRECT ANSWER
-cos 50°
EDDIE SAYS
The 230° line makes an angle of 50° with the horizontal.
230° is in the 3rd quadrant where cos is negative.
  • Question 5

Change the following to an equivalent trigonometric function of an angle less than 90º:

 

tan 230º

CORRECT ANSWER
tan 50°
EDDIE SAYS
The 230° line makes an angle of 50° with the horizontal.
230° is in the 3rd quadrant where tan is positive.
  • Question 6

Change the following to an equivalent trigonometric function of an angle less than 90º:

 

sin 280º

CORRECT ANSWER
-sin 80°
EDDIE SAYS
The 280° line makes an angle of 80° with the horizontal.
280° is in the 4th quadrant where sin is negative.
  • Question 7

Change the following to an equivalent trigonometric function of an angle less than 90º:

 

cos 280º

CORRECT ANSWER
cos 80°
EDDIE SAYS
The 280° line makes an angle of 80° with the horizontal.
280° is in the 4th quadrant where cos is positive.
  • Question 8

Change the following to an equivalent trigonometric function of an angle less than 90º:

 

cos 132º

CORRECT ANSWER
-cos 48°
EDDIE SAYS
The 132° line makes an angle of 48° with the horizontal.
132° is in the 2nd quadrant where cos is negative.
  • Question 9

Change the following to an equivalent trigonometric function of an angle less than 90º:

 

sin 365º

CORRECT ANSWER
sin 5°
EDDIE SAYS
The 365° line makes an angle of 5° with the horizontal.
365° is in the 1st quadrant where sin is positive.
  • Question 10

Change the following to an equivalent trigonometric function of an angle less than 90º:

 

tan 580º

CORRECT ANSWER
tan 40°
EDDIE SAYS
The 580° line is the same as the 580 - 360 = 220º line which makes an angle of 40° with the horizontal.
580° and 220° are in the 3rd quadrant where tan is positive.
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