45˚- 45˚- 90˚ TrianglesDrawing the diagonal of a square separates it

into two 45˚- 45˚- 90˚ triangles.Because it’s a square, the

legs of the triangle are equal (that makes it isosceles)

Because it’s a right triangle, we can use the Pythagorean Theorem to find the length of the diagonal.

45˚- 45˚- 90˚ TrianglesIf the sides are of length x,

then the diagonal of the square (the hypotenuse of the triangle) is:x2 + x2 = c2

2x2 = c2

= c = c

To get the hypotenuse, simply multiply the side length by

x

x

22x2x

2x

2

45˚- 45˚- 90˚ TrianglesExample: An official

baseball diamond is a square with sides 90 ft long. How long is it fromhome plate to secondbase?

Answer: 90 2

45˚- 45˚- 90˚ TrianglesSometimes, you’ll be given the hypotenuse

and asked to find the side length.In these cases, you will divide by . The thing

to remember here is that you will have to rationalize your answer.

2

45˚- 45˚- 90˚ TrianglesExample of finding sides given hypotenuse.If PQR is an isosceles right triangle, and the

measure of the hypotenuse is 12, find s.

12 12 22

62

22 2

45˚- 45˚- 90˚ TrianglesYOUR TURN

Find the missing measures. Write all radicals in simplest form.

x = 8

8 2y

x = 10

10 2y

45˚- 45˚- 90˚ TrianglesYOUR TURN

Find the missing measures. Write all radicals in simplest form.

x = 3 9 2x y = 3 9 2y

45˚- 45˚- 90˚ TrianglesAssignment

Worksheet #13-2

30˚- 60˚- 90˚ TrianglesFor 45/45/90 right triangles, we separated a

square into two equal parts.For 30/60/90 triangles, we separate

equilateral triangles into two equal parts.

60˚ 60˚

60˚

30˚- 60˚- 90˚ TrianglesProof of the ratios up on the board.

The key to 30˚- 60˚- 90˚ triangles is to find the shortest side (the side opposite the 30˚)

3x

x2x

30˚

60˚

30˚- 60˚- 90˚ TrianglesExample

In ABC, b = 7. Find a and c. Write in simplest radical form. b is the shortest side The hypotenuse (c) is 2x the

shortest leg c = 14

The longer leg (a) is timeslarger than the shortest leg a =

3

7 3

30˚- 60˚- 90˚ TrianglesExample

In ABC, c = 18. Find a and b. Write in simplest radical form. c is the hypotenuse The hypotenuse is 2x the

shortest leg (b) b = 9

The longer leg (a) is timeslarger than the shortest leg a =

3

9 3

30˚- 60˚- 90˚ TrianglesYOUR TURN

In ABC, b = 8. Find a and c. a = c = 16

In ABC, c = 10. Find a and b. a = b = 5

5 3

8 3

30˚- 60˚- 90˚ TrianglesExample

In ABC, a = 12. Find b and c. Write in simplest radical form. The longer leg is times

larger than the shortest leg (b) By dividing by , we can find the

short leg. Then we have to rationalize

b =

The hypotenuse (c) is 2x the shortest leg c =

3

12 12 334 3

3 3 3

3

8 3

30˚- 60˚- 90˚ TrianglesYour Turn

In ABC, a = 8. Find b and c. Write in simplest radical form.

b =

c =

8 3

3

16 3

3

30˚- 60˚- 90˚ TrianglesAssignment

Worksheet 13-3

Essential Question: What do the sine and cosine of an angle represent on the unit circle?

13-2: Angles and the Unit CircleAn angle is in standard position when the vertex

is at the origin and one ray is on the positive x-axis (one angle pointing straight right).

The ray on the x-axis (pointing right) is the initial side of the angle.

The other ray is the terminal side of the angle.

13-2: Angles and the Unit CircleTo measure an angle in standard position, find the

amount of rotation from the initial side to the terminal side.

One full rotation contains 360˚.How many degrees are in one quarter of a rotation?In one half a rotation?In three quarters of a rotation?

The angle measures 20˚ more thana straight angle of 180˚.

Since 180 + 20 = 200, the angle is 200˚

13-2: Angles and the Unit CircleThe measure of an angle is positive when the

rotation is counterclockwise. The measure is negative when the rotation is clockwise.

13-2: Angles and the Unit CircleFind the measure of each angle in standard

position-315˚ -135˚ 240˚

115˚ -110˚ -340˚

13-2: Angles and the Unit CircleTwo angles in standard position are

coterminal if they have the same terminal side

To find another coterminal angle, simply add or subtract a revolution (360˚)

There are an infinite number of angles that can be coterminal. Simply keep adding (or subtracting) revolutions…

13-2: Angles and the Unit CircleFind another angle coterminal with 198˚ by

adding one full rotation.558˚

Are angles with measures of 40˚ and 680˚ coterminal? ExplainNo, 40˚ + 360 = 400˚ is coterminalAnd so is 400˚ + 360˚ = 760˚But nothing else in between is coterminal

13-2: Angles and the Unit CircleAssignment

Worksheet Practice 13-221 – 49 (odds)69 – 71 (all)

Essential Question: What do the sine and cosine of an angle represent on the unit circle?

13-2: Angles and the Unit CircleUnit Circle: A circle with a radius of 1 unit

Θ: The Greek letter “theta”, used for angle measurements

13-2: Angles and the Unit CircleCosine of Θ (cos Θ): The x-coordinate where

an angle intersects the unit circleSine of Θ (sin Θ): The y-coordinate where an

angle intersects the unit circle

P(cos Θ, sin Θ)

13-2: Angles and the Unit CircleExample: Find the cosine and sine of 60˚

Draw a unit circleDraw the angle indicatedConstruct a right triangleUse the rule for 45/45/90 or 30/60/90 triangles to

determine the sine and cosine

60˚

1sin x

cos x

1cos60 0.5

2

3sin 60 0.8660

2

13-2: Angles and the Unit CircleYour Turn

Find the cosine and sine of 45˚

Find the cosine and sine of 30˚

2cos45 0.7071

2 2

sin 45 0.70712

1sin30 0.5

2 3

cos30 0.86602

13-2: Angles and the Unit CircleWhat do we do about angles greater than 90˚?What do we do about negative angles?

The same thing… exceptBe aware of negative values.If the triangle is drawn to the left, the cosine

value is negativeIf the triangle is drawn to the bottom, the sine

value is negativeThere’s a pneumonic device we will get to after

this example

13-2: Angles and the Unit CircleExample: Find the cosine and sine of -120˚

Draw a unit circleDraw the angle indicatedConstruct a right triangleUse the rule for 45/45/90 or 30/60/90 triangles to

determine the sine and cosine

60˚

1

cos x

sin x

1cos ( 120) 0.5

2

3sin ( 120) 0.8660

2

13-2: Angles and the Unit CircleAnother trick on when sin/cos should be

positive or negativeASTC (All Students Take Calculus)

Tells you (by quadrant) which values are positive We will get to tangent eventually, but it’s worth

discussing nowQuad I

Allvalues

Quad IIsin

only

Quad IIItanonly

Quad IVcosonly

13-2: Angles and the Unit CircleAssignment

Page 722Problems 21-28All problems

Sketches are required for credit