Trigonometry About the Four Quadrants Ratios for 0

Trigonometry About the Four Quadrants Ratios for 0 and 90

Trigonometry The Four Quadrants Co-ordinates in the First Quadrant Trig Ratios in the First Quadrant Co-ordinates as angle θ decreases Trig Ratios of 0° Co-ordinates as angle θ increases Trig Ratios of 90°

The Four Quadrants 90 0 180 360 270

THE FOUR QUADRANTS. Take a pizza and cut it into four equal pieces. Second Quarter Third Quarter Y axis Each piece is one quarter of the whole pizza. First Quarter Fourth Quarter X axis Now take the pizza away.

THE FOUR QUADRANTS. In trigonometry we Y axis work with right angled triangles and the four quarters. Second First Quarter We call the quarters quadrants. Third Quarter O Fourth Quarter X axis

THE FOUR QUADRANTS. In trigonometry we Y axis work with right angled triangles and the four quarters. Second First r y Quadrant O We call the quarters quadrants. Third Quadrant Fourth Quadrant X axis

THE FOUR QUADRANTS. One full revolution is 360. 90 One quarter revolution is 90. 4 x 90 =360 180 O 270 0 X axis 360

THE FOUR QUADRANTS. All the work we will do in this lesson is in the first quadrant. 180 90 O 270 We will work from 0 to 90. 0 X axis 360

Well Done!

Co-ordinates in the First Quadrant A(x, y) y θ x

The Basic Trig Triangle § In triangle ABC, AB = 5 units, BC = 3 units and AC = 4 units. § How would you prove that triangle ABC is right angled? A Did someone say Pythagoras? B 5 4 3 C

The Basic Trig Triangle § Pythagoras’s theorem states: § The square on the hypotenuse equals the sum of the squares on the other two sides. In this triangle: A B 5 4 Is AB 2 = AC 2 + BC 2 ? 52 = 4 2 + 3 2 ABC is 25 = 16 + 9 TRUE This means right angled. 3 C

Draw the basic 3, 4, 5 triangle Draw a circle around it The radius is the length of the hypotenuse 5 4 3

The centre of the circle is at the letter O Point O is called the origin Y axis Draw axes through the origin What are the co-ordinates of this point? 5 O 4 3 X axis

How far do we move along the X and Y axis to get to the point? Y axis What are the co-ordinates of this point? (4, 3) 5 3 4 O X axis Move 4 along X axis. Move 3 along Y axis Co-ordinates of the point are: (4, 3)

How far do we move along the X and Y axis to get to the point? Y axis What are the co-ordinates (3, 4)of this point? 5 O 4 3 X axis Move 3 along X axis. Move 4 along Y axis Co-ordinates of the point are: (3, 4)

The co-ordinates of point A are (4, 3) What are the values of x and y in the diagram? Y axis A(4 , 3) y O x X axis x=4 x is the distance along the X axis to get to the point.

The co-ordinates of point A are (4, 3) What are the values of x and y in the diagram? Y axis A(4 , 3) y O x X axis y=3 y is the distance along the Y axis to get to the point.

The co-ordinates of point A are (3, 4) What are the values of x and y in the diagram? Y axis A(3 , 4) y O x X axis x=3 x is the distance along the X axis to get to the point.

The co-ordinates of point A are (3, 4) What are the values of x and y in the diagram? Y axis A(3 , 4) y O x X axis y=4 y is the distance along the Y axis to get to the point.

Now you know what co-ordinates are. Let’s look at a general case. Y axis What are the co-ordinates of this point? r x O y X axis

How far do we move along the X and Y axis to get to the point? Y axis What are the co-ordinates of this point? (x, y) r y x X axis O Move x along X axis. Move y along Y axis Co-ordinates of the point are: (x, y)

Well Done!

Trig Ratios in the First Quadrant Tan Sin θ Cos θ Tangent Sine θ Cosine θ

In the diagram, can you work out the trig ratios for angle in terms of the sides of the triangle? Y axis We need to label the opposite and adjacent sides, (x, y) and the r hypotenuse. y x X axis O Co-ordinates of the point are: (x, y)

In the diagram, can you work out the trig ratios for angle in terms of the sides of the triangle? Y axis We need to label the opposite and adjacent sides, (x, y) and the r hypotenuse. O y O x X axis

In the diagram, can you work out the trig ratios for angle in terms of the sides of the triangle? Y axis We need to label the opposite and adjacent sides, (x, y) and the H r hypotenuse. O y Tan = = A x O y Sin = = H r O x A X axis A x Cos = = H r

Now you know how to write down the trig ratios of angle in terms of the sides x, y and r of a right angled triangle. Y axis (x, y) r O y Tan = = A x y O = Sin = H r O y x X axis A x Cos = = H r

Well Done!

Co-ordinates as Angle θ gets smaller A(x, y)

What happens to x and y as angle decreases? Start with point A(x, y) in the first quadrant. 90 A(x, y) What happens when angle gets smaller? r y x O Will the radius r change? Yes No 0 X axis

Oops! When you draw a circle the compass width stays the same. Click to try again.

Well Done!

What happens to x and y as angle decreases? Watch what happens when changes. 90 A(x, y) r O y x 0 X axis

What happens to x and y as angle decreases? Watch what happens when changes. 90 A(x, y) r O x y 0 X axis

What happens to x and y as angle decreases? Watch what happens when changes. 90 A(x, y) r O x y 0 X axis

What happens to x and y as angle decreases? Watch what happens when changes. 90 r O x A(x, y) y 0 X axis

What happens to x and y as angle decreases? Watch what happens when changes. 90 r O x A(x, y) y 0 X axis

What happens to x and y as angle decreases? Watch what happens when changes. 90 O r x A(x, y) y 0 X axis

What happens to x and y as angle decreases? Watch what happens when changes. 90 A(x, y) O When = 0 , how big is side y? r x y 0 y=r y=0 y=x X axis

Oops! Look at the slides again and see what happens to the length of y. Click to try again.

Well Done!

What happens to x and y as angle decreases? Watch what happens when changes. 90 A(x, y) O When = 0 , how big is side x? r x y 0 x=r x=0 x=y X axis

Oops! Look at the slides again and see what happens to the length of x. Click to try again.

Well Done!

The Trig Ratios of 0° 0 ° A(x, y)

The Trigonometric ratios of 0 What is the value of the following trig ratio? 90 O Opp Tan = Adj r x A(x, y) y 0 X axis 0 r 1

Oops! y Opp Tan = = x Adj 0 = = 0 r Click to try again.

Well Done!

The Trigonometric ratios of 0 What is the value of the following trig ratio? 90 O Opp Sin = Hyp r x A(x, y) y 0 X axis 0 r 1

Oops! y Opp Sin = = r Hyp 0 = = 0 r Click to try again.

Well Done!

The Trigonometric ratios of 0 What is the value of the following trig ratio? 90 O Adj Cos = Hyp r x A(x, y) y 0 X axis 0 r 1

Oops! x Adj Cos = = r Hyp r = = 1 r Click to try again.

Well Done!

Co-ordinates as Angle θ gets Bigger A(x, y)

What happens to x and y as angle increases? Start with point A(x, y) in the first quadrant. 90 A(x, y) What happens when angle gets bigger? r y x O Will the radius r change? Yes No 0 X axis

Oops! When you draw a circle the compass width stays the same. Click to try again.

Well Done!

What happens to x and y as angle increases? Watch what happens when changes. 90 A(x, y) r O y x 0 X axis

What happens to x and y as angle increases? Watch what happens when changes. 90 A(x, y) r O y x 0 X axis

What happens to x and y as angle increases? Watch what happens when changes. 90 r O y x A(x, y) 0 X axis

What happens to x and y as angle increases? Watch what happens when changes. 90 r O x A(x, y) y 0 X axis

What happens to x and y as angle increases? Watch what happens when changes. 90 A(x, y) r O x y 0 X axis

What happens to x and y as angle increases? Watch what happens when changes. 90 A(x, y) r y Ox When = 90 , how big is side y? 0 X axis y=r y=0 y=x

Oops! Look at the slides again and see what happens to the length of y. Click to try again.

Well Done!

What happens to x and y as angle increases? Watch what happens when changes. 90 A(x, y) r y Ox When = 90 , how big is side x? 0 X axis x=r x=0 x=y

Oops! Look at the slides again and see what happens to the length of x. Click to try again.

Well Done!

The Trig Ratios of 90° 90 r y x 0 X axis

The Trig Ratios of 90° What is the value of the following trig ratio? 90 A(x, y) r y Ox Opp Tan = Adj 0 X axis 0 1 undefined

Oops! y r Opp Tan = = x Adj 0 You can’t divide by 0. Click to try again. ∞

Well Done!

The Trig Ratios of 90° What is the value of the following trig ratio? 90 A(x, y) r y Ox Opp Sin = Hyp 0 X axis 0 1 undefined

Oops! y Opp Sin = = r Hyp r = = 1 r Click to try again.

Well Done!

The Trig Ratios of 90° What is the value of the following trig ratio? 90 A(x, y) r y Ox Adj Cos = Hyp 0 X axis 0 1 undefined

Oops! x Adj Cos = = r Hyp 0 = = 0 r Click to try again.

Well Done!