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Mathematics Teaching Self-evaluation Tools

You are viewing a limited version of the NCETM’s self-evaluation tools. Any answers you save during this session will be removed after seven days. Log in or sign up to view, and use, the full version of the tools.

For the following questions, select the statement which most accurately matches your level of confidence (1 is not confident and 4 is very confident) or choose from the alternatives detailed in the question. You do not have to answer all questions. Your answers will be saved so you can exit and come back to your self-evaluation at any time. Click Save and Results to view the next steps for questions you have answered.

1.

How confident are you that you can use the cosine rule to:

1

2

3

4

a.

find missing lengths and angles of a non-right-angled triangle?

Cosine rule: in any triangle ABC with lengths a, b, c:

c^{2} = a^{2} + b^{2} – 2ab cos C

The labelling of the triangle is arbitrary except that a is the length opposite vertex A, b is the length opposite vertex B and so on.

The cosine rule can also be written as:

a^{2} = b^{2} + c^{2} – 2bc cos A or b^{2} = a^{2} + c^{2} – 2ac cos B

Example 1: IntriangleABC, AC = 17cm, BC = 9cm and angleACB = 20°. Calculate lengthAB.

Using the cosine rule gives

AB^{2} = 9^{2} + 17^{2} – 2 × 9 × 17 cos 20° AB^{2} = 82.45 AB = 9.08 cm which is the length to 2dp.

Note that the cosine rule works for any triangle including right-angled triangles. When you have a right-angled triangle the cosine rule defaults to Pythagoras’ Theorem.

Example 2: In triangle ABC: a = 12 cm b = 6 cm c = 8 cm

Find cos B, giving your answer in exact form.

From the cosine rule: b^{2} = a^{2} + c^{2} – 2ac cos B

This gives 6^{2} = 12^{2} + 8^{2} – 2 × 12 × 8 cos B

Therefore 36 = 144 + 64 – 182 cos B 36 = 208 – 192 cos B –172 = –192 cos B cos B =

172

192

=

43

48

2.

How confident are you that you can use the sine rule to:

1

2

3

4

a.

find missing lengths and angles of a non-right-angled triangle?

Sine rule: in any triangle ABC with lengths a, b, c

a

sin A

=

b

sin B

=

c

sin C

.

Example: In triangle ABC, A = 30º, a = 4 cm and b = 7 cm. Find the other angles in the triangle using the sine rule.

Using the sine rule gives:

a

sin A

=

b

sin B

or

sin A

a

=

sin B

b

So B = 61º or 119º

If B = 61º then C = 89º

If B = 119º then C = 31º.

3.

How confident are you that you can:

1

2

3

4

a.

choose the appropriate rule to apply to the problem?

If you are given all three sides then using the sine rule will lead you to trying to solve equations with two unknowns. For example, if a = 3, b = 6, c = 5, trying to find angle A using the sine rule leads to or , neither of which can be solved.

Similarly if you are given two sides and the included angle and use the sine rule, you are again faced with trying to solve an equation with two unknowns.

4.

How confident are you that you can introduce and explain the concept of radians:

1

2

3

4

a.

radians as an alternative to degrees as a measure of angle?

The use of degrees to measure angles is arbitrary but convenient. A full turn is defined as having an angle of 360 degrees. The choice of 360 is due to the ease with which we can calculate angles of common fractions of a circle. For example, half a circle is 180 degrees, a third of a circle contains 120 degrees and a quarter circle contains 90 degrees etc.

Other measures of angle have also been defined for convenience. A full turn has been defined as containing 400 gradians so that a right-angle has a convenient 100 gradians. The gradian is often used by architects and engineers.

The binary gradian (or brad) has been defined so that a full turn has 256 brads. This is convenient for computer programmers who can use a single byte of memory to store an angle.

In the same way, the radian has been defined so that there are 2π radians in a full turn.

If you have a sector of a circle with radius r and the arc length is also r, the angle of the sector is defined as 1 radian.

Examples: Sort the following into 3 groups:

(a) ones that equal 0 (b) ones that equal 1 (c) ones that equal -1

Why is tan

π

2

not on the list?

sin π

cos 2π

tan π

cos 0

sin 2π

tan 3π

sin 0

sin

π

2

tan 0

cos

3π

2

cos π

tan

π

4

tan 2π

cos

π

2

sin

3π

2

sin 3π

cos 4π

tan

3π

4

(a) Equal to 0:

sin π

tan π

sin 2π

cos 0

tan 0

cos

3π

2

tan 2π

cos

π

2

sin 3π

tan 3π

(b) Equal to 1:

sin

π

2

tan

π

4

cos 0

cos 4π

cos 2π

(c) Equal to -1:

cos π

sin

3π

2

tan

3π

4

tan

π

2

is not on the list because its value it infinite.

5.

How confident are you that you can use radian measure to:

1

2

3

4

a.

find and apply a formula for arc length?

If θ is measured in radians, the formula for arc length L is:

Example : If the radius is 3 cm, and the length of the arc is 10 cm, find θ.

10 = 3 x θ

So θ = 3.33 radians.

Example 2: A sector is cut from a circle with angle 0.8 radians at the centre. If the perimeter of the sector is 28 cm, find the radius of the circle.

Arc length = r × θ = 0.8r Perimeter = r + r + 0.8r = 2.8r

So 2.8r = 28 ⇒ r = 10 cm.

6.

How confident are you that you can use radian measure to find and apply:

1

2

3

4

a.

a formula for area of a sector of a circle?

If θ is measured in radians, the formula for the area A of the sector is:

Example: If θ = 2 radians, find the area of the shaded section in this diagram.

Shaded area = area of sector OAB – Area of triangle OAB.

Area of sector OAB = 0.5 × 9 × 2 = 9 cm^{2}

Triangle OAB is isosceles with base 6 sin 1 and height 3 cos 1. Area of triangle is 0.5 x 6 sin 1 x 3 cos 1 = 4.1 cm^{2}

Shaded area = 9 - 4.1 = 4.9 cm^{2}.

7.

How confident are you that you know and can explain the properties of:

1

2

3

4

a.

the sine function?

Properties of the sine function:

y = sin x is periodic i.e. the shape of the curve repeats after an interval.

The period of y = sin x is 360° or 2π.

The graph has rotational summetry about the origin of order 2 which means that sin(-x) = -(sin x).

The maximum value of y = sin x is 1 and the minimum is -1.

b.

the cosine function?

Properties of the cosine function.

y = cos x is periodic with period 360° or 2π.

The graph has reflection symmetry in the y-axis which means that cos x = cos (-x).

The maximum value of y = cos x is 1 and the minimum value is -1.

c.

the tangent function?

Properties of the tangent Function

y = tan x is a periodic function with period 180° or π.

It has rotational symmetry about the origin of order 2 and -(tan x) = tan(-x)

The tangent function has no maximum or minimum values but it is undefined when x = ±90°, ±270°, ±450°, …

8.

How confident are you that you can explain:

1

2

3

4

a.

why sin θ / cos θ = tan θ and use this to solve simple trigonometric equations?

In the right-angled triangle ,

Example Solve 4 sin θ – 3 cos θ = 0 for 0 ≤ θ ≤ 360°.

This is the principal value and was found by using a calculator.

As tan θ has a period of 180° there is another solution to at 36.9 + 180 = 216.9°.

Therefore θ = 36.9°, 216.9° for 0 ≤ θ ≤ 360°.

9.

How confident are you that you can explain why:

1

2

3

4

a.

sin² θ + cos² θ = 1 and use this to solve simple trigonometric equations?

Consider a right-angled triangle with hypotenuse 1. Then O = sin θ and A = cos θ.

By Pythagoras’ theorem: sin^{2} θ + cos^{2} θ = 1

Example 1: Solve 2cos^{2} θ = 3(1 – sin θ) for 0 ≤ θ ≤ 360°.

Replacing cos^{2} θ with (1 – sin^{2} θ) gives 2(1 – sin^{2} θ) = 3(1 – sin θ) 2 – 2sin^{2} θ = 3 – 3sin θ 2sin^{2} θ – 3sin θ + 1 = 0 which is a quadratic equation in sin θ.

Solving gives sin θ = 1 or sin θ =

1

2

So θ = 90° or θ = 30°.

These are the principal values. There is another solution which occurs at 180 – 30 = 150°.

Therefore the solutions are θ = 30°, 90°, 150°.

Example 2: If sin θ =

1

4

find two possible values of cos θ, leaving your answers in exact form.