Grade 9

Lesson 4-9: Solving polynomial inequalities

Posted on Updated on

In this section we will be solving (single) inequalities that involve polynomials of degree at least two. Since it’s easier to see the process as we work an example let’s do that.  As with the linear inequalities, we are looking for all the values of the variable that will make the inequality true.  This means that our solution will almost certainly involve inequalities as well.  The process that we’re going to go through will give the answers in that form.

How To Solve Polynomial Inequalities?

Here are the steps to solve polynomial inequalities:

  1. Make your inequality looks like P(x) < 0 (the inequality symbol can also be > or)
  2. Find the zeros of P(x) (by factoring it)
  3. Plot the zeros on the number line
  4. Pick a test point on each interval and plugin it to P(x)
  5. If after plugin you get it true then the interval is a part of the solution otherwise it’s not

The following examples will guide you step by step to solve the polynomial inequalities:

Example 1: Solve x² – 10 < 3x

step1: Make your inequality looks like P(x) < 0 (the inequality symbol can also be > or)

x² – 3x -10 < 0

step2: Find the zeros of P(x) (by factoring it)

(x + 2)(x – 5) < 0

So, the zeros of x are {-2 ,  5}

step3: Plot the zeros on the number line

step4: Pick a test point on each interval and plugin it to P(x)


step5: If after plugin you get it true then the interval is a part of the solution otherwise it’s not

So the solution is between -2 and 5,   -2 < x < 5

 

Example 2: Solve x²( + 4x – 12) ≤ 0

step1: Make your inequality looks like P(x) < 0 (the inequality symbol can also be > or)

x²( + 4x – 12) ≤ 0

step2: Find the zeros of P(x) (by factoring it)

x²(x + 6)(x – 2) ≤ 0

So, the zeros of x are {-6, 0 ,  2}

step3: Plot the zeros on the number line

step4: Pick a test point on each interval and plugin it to P(x)

step5: If after plugin you get it true then the interval is a part of the solution otherwise it’s not

So the solution is between -6 and 2,   -6 ≤ 2

 

Practice 7-4: Trigonometry

Posted on Updated on

Answer the following questions

1.

which ratio would you use to find angle Φ?


Choose one:

Cosine

Sine

tangent


2
.

which ratio would you use to find angle α?


Choose one:

Cosine

Sine

Tangent

3.

The ratio 5/8 is ….?


Choose one:

Cos (Φ)

Cos (θ)

Tan (θ)


4
.

what is the value of cosθ?


Choose one:

4/5

3/5

3/4


5
.

What is the value of tan Φ?


Choose one:

8/17

15/17

15/8

6.

What is the size of angle x?


Choose one:

44.4°

55.0°

44.6°


7
.

The diagram shows a vertical tree of height 22 ft. From the point A (on level ground) the horizontal distance to the base B of the tree is 30 ft. What is the angle of elevation from A to the top of the tree C?


Choose one:

42.8°

47.2°

36.3°


8
.

What is the size of angle x?


Choose one:

53.1°

51.3°

38.3°


9
.

What is the size of angle x?


Choose one:

52.0°

51.3°

32.0°

Random Trigonometry

Lots of random trigonometry problems

 

Back to Lesson 7-4: Trgonometry

Lesson 7-4: Trigonometry

Posted on Updated on

Trigonometry (from Greek trigōnon “triangle” + metron “measure”) is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides.

Sine, Cosine and Tangent

The three main functions in trigonometry are Sine, Cosine and Tangent.  They are often shortened to sin, cos and tan. they are based on a Right-Angled Triangle, where they help us finding an unknown angle or side in a right-angled triangle.

Before getting stuck into these functions, it helps to give a name to each side of a right triangle:

  • “Opposite” is the side that faces angle θ
  • “Adjacent” is the side (next to) angle θ
  • “Hypotenuse” is the longest side and it always faces the right angle

How to calculate them

For a right triangle with an angle θ, the functions Sine, Cosine and Tangent are calculated this way:

How to Remember them?

well “Sohcahtoa” may be easy for you to remember… 

Soh…
Sine = Opposite / Hypotenuse
…cah…
Cosine = Adjacent / Hypotenuse
…toa
Tangent = Opposite / Adjacent

 

Why are these functions important?

  • Because they let you work out angles when you know sides
  • And they let you work out sides when you know angles

 Example: A 5ft ladder leans against a wall as shown.

What is the angle between the ladder and the wall?

The answer can be found using: Sine, Cosine or Tangent!

But which one to use?

Remember: we have a special phrase “SOHCAHTOA” to help us, and we use it like this:

Step 1: find the names of the two sides you know

In our ladder example we know the length of:

  • the side Opposite the angle “x” (2.5 ft)
  • the long side, called the “Hypotenuse” (5 ft)

Step 2: now use the first letters of those two sides (Opposite and Hypotenuse) and the phrase “SOHCAHTOA” to find which one of Sine, Cosine or Tangent to use:

SOH…
Sine: sin(θ) = Opposite / Hypotenuse
…CAH…
Cosine: cos(θ) = Adjacent / Hypotenuse
…TOA
Tangent: tan(θ) = Opposite / Adjacent

In our example that is Opposite and Hypotenuse, and that gives us “SOHcahtoa”, which tells us we need to use Sine.

Step 3: Put our values into the Sine equation:

Sin (x) = Opposite / Hypotenuse = 2.5 / 5 = 0.5

Step 4: Now solve that equation!

sin (x) = 0.5

Next (trust me for the moment) we can re-arrange that into this:

x = sin-1 (0.5)

And then get our calculator, key in 0.5 and use the sin-1 button to get the answer:

x = 30°

But what is the meaning of sin-1 … ?

Well, the Sine function “sin” takes an angle and gives us the ratio “opposite/hypotenuse”,

But in this case we know the ratio “opposite/hypotenuse” but want to know the angle.
So we want to go backwards. That is why we we use sin-1, which means “inverse sine”.

Example:

  • Sine Function: sin(30°) = 0.5
  • Inverse Sine Function: sin-1(0.5) = 30°

Examples:

Let’s look at a couple more examples:

1 ) Find the size of the angle of elevation (x) of the plane from point A on the ground.

Step 1 The two sides we know are Opposite (300) and Adjacent (400).

Step 2 SOHCAHTOA tells us we must use Tangent.

Step 3 Use your calculator to calculate Opposite/Adjacent

Step 4 Find the angle from your calculator using tan-1

Tan x° = opposite/adjacent

Tan x° = 300/400

Tan x° =  0.75

x° = tan-1 (0.75)

x° = 36.9°

2 ) Find the size of angle a°


Step 1 The two sides we know are Adjacent (6,750) and Hypotenuse (8,100).

Step 2 SOHCAHTOA tells us we must use Cosine.

Step 3 Use your calculator to calculate Adjacent / Hypotenuse

Step 4 Find the angle from your calculator using cos-1

cos a° = 6,750/8,100

cos a° = 0.8333

a° = cos-1(0.8333)
a° =33.6°

Practice 8-1: Angles of Polygons

Posted on Updated on

Answer the following questions

1.

What is the sum of the exterior angles of an octagon?

Choose one:

480°

360°

720°


2
.

What is the size of one exterior angle of a regular decagon (ten-sided polygon)?

Choose one:

24°

18°

36°

3.

One exterior angle of a regular polygon is 20°. How many sides does it have?

Choose one:

15

18

16


4
.

What is the sum of the interior angles of a regular dodecagon (12-sided polygon)?

Choose one:

2160°

1080°

1800°


5
.

The diagram shows a hexagon. What is the size of the angle x°?.


Choose one:

110°

130°

120°

6.

What is the size of one interior angle of a regular nonagon (nine-sided polygon)?

Choose one:

140°

126°

40°


7
.

Each of the interior angles of a regular polygon is 150°. How many sides does the polygon have?

Choose one:

11

14

12


Back to Lesson 8-1: angles of polygons