using the cartesian plane, you can find the trigonometric ratios for angles with measures greater...

Using the Cartesian plane, you can find the trigonometricratios for angles with measures greater than 900 or lessthan 00. Angles on the Cartesian plane are called rotational angles.An angle is in standard position when the initial arm is onthe positive x-axis and the vertex is at (0, 0).

Initial Arm

TerminalArm

Vertex (0, 0)

Angles in Standard Position

An angle is positive when the rotation is counterclockwise.

An angle is negative when the rotation is clockwise.

Quadrant I

Quadrant II

Quadrant III

Quadrant IV


Principal Angle

Reference Angle

is measured from the positive x-axis tothe terminal arm.

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is measured in a counterclockwise direction, therefore is always positive.

is always less than 3600.



is the acute angle between the terminal arm and the closest x-axis.

is measured in a counterclockwise direction, therefore is always positive.

is always less than 900.





PrincipalAngleReference

Angle

PrincipalAngle

ReferenceAngle

PrincipalAngle

ReferenceAngle


Sketch the following angles and list the reference and principal angles.

A) 1200 B) -1200 C) 800 D) 2400

Principal Angle

Principal Angle

Principal Angle

Principal Angle

ReferenceAngle

ReferenceAngle

ReferenceAngle

ReferenceAngle

1200 2400 800 2400

600600 800 600

Finding the Reference and Principal Angles

Choose a point (x, y) on the terminal arm and calculate the primary trig ratios.

P(x, y)

x

yr

q

sin y

r

cos x

r

tan y

xr2 = x2 + y2

x2 = r2 - y2

y2 = r2 - x2

Finding the Trig Ratios of an Angle in Standard Position

P(x, y)

y r

q

xNote that x

is a negative number

r2 = (x)2 + y2

(x)2 = r2 - y2

y2 = r2 - (x)2


r

xcosθ

r

yθsin

x

yθtan

Remember that in quadrant II, x is negative so cosine and tangent willbe negative.

The point P(3, 4) is on the terminal arm of q . List the trig ratios and find q .

q

P(3, 4)

3

4

r2 = x2 + y2

= 32 + 42

= 9 + 16 = 25 r = 5

5sin

4

5

cos 3

5

tan 4

3

q = 530


P(-3, 4)

q

The point P(-3, 4) is on the terminal arm of q . List the trig ratios and find q .

-3

4

r2 = x2 + y2

= (-3)2 + (4)2

= 9 + 16 = 25 r = 5

sin 4

5cos

3

5

tan 4

3q ref= 530

5

ReferenceAngle

1800 - 530 = 1270

Principal Angle

q = 1270


P(-2, 3)

q-2

3

r2 = x2 + y2

= (-2)2 + (3)2

= 4 + 9 = 13 r = √ 13

sin 3

13cos

2

13

tan 3

2

13

q ref= 560

Reference Angle !!from your calculator

1800 - 560 = 1240

q = 1240

Principal Angle

The point P(-2, 3) is on the terminal arm of q . List the trig ratios and find q .


Related angles are principal angles that have the same reference angles. These angles will also have the same trig ratios. The signs of the ratio may differ depending on the quadrant that they are in.

300300

300

sin 300 = 0.5

PA = 300

PA = 1500

PA = 2100

Sin 1500 = 0.5 sin 2100 = -0.5

Related Angles

Using the ASTC Rule

Cosine

AllSine

Tangent

q1800 - q

Evaluate to four decimal places.

A) sin 1370 = 0.6820

B) cos 1420 = -0.7880C) tan 1580 = -0.4040

Find angle A, to the nearest degree: 00 ≤ A < 1800

sin A = 0.3415 200 RA 200 1600

cos A = -0.4318 640 RA 1160

tan A = -1.4132 550 RA 1250

cos A = 0.6328 510 RA 510

I II

Cosine

AllSine

Tangent

q1800 - q

Find angle A, to the nearest degree: 00 ≤ A < 3600

sin A = 0.5632

cos A = -0.7542

tan A = -1.5643

cos A = 0.5986

1800 + q 3600 - q

sin A = -0.8667

tan A = 0.5965

RA Quadrants

I II

II III

II IV

I IV

III IV

I III

340

410

570

530

600

310

340 1460

1390 2210

1230 3030

530 3070

2400 3000

310 2110

Using the ASTC Rule (All Students Take Calculus)

using the cartesian plane, you can find the trigonometric ratios for angles with measures greater...

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