C2 - Degrees Trig

Question 1

1.1.1 What is the Cosine Rule?

Answer 1

a^2 = b^2 + c^2 - 2bc * Cos(A)

Alternatively:

Cos(A) = (b^2 + c^2 - a^2 )/2bc

Question 2

1.1.2 What is the importance of the labelled sides?

Answer 2

The lower case letters represent the sides of the triangle, and their corresponding upper case letters represent the angle OPPOSITE the side.

Take care of this when using the formula. If you input the wrong sides, it will result in a wrong answer (that being said, side b and c are interchangeable).

Question 3

1.1.3 What is the Sine rule?

Answer 3

a/Sin(A) = b/Sin(B) (=c/Sin(C))

Can alternatively be written as the Sin function over the corresponding side of the angle.

Question 4

1.1.4 What is the standard formula for the area of a triangle?

Answer 4

1/2 * a * b * sin(C)

Take care with the sides again. The side opposite the angle for the sine function should NOT be used in the calculation.

Question 5

1.1.5 Extra: why is the area for the right angle triangle only 1/2 * b * h?

Answer 5

Because sin(90) = 1, and is the equation is 1/2 * a * b * 1, which is just 1/2 * a * b.

Question 6

1.2.1 What is the trigonometrical identity derived from Pythagoras’ theorem?

Answer 6

(Sin(x))^2 + (cos(x))^ 2 == 1

It is easy to rearrange for solutions of equations (later topic). It’s not particularly useful in other cases, however.

Proof:

In a triangle a^2 + b^2 = c^2

(a^2/c^2) + (b^2/c^2) = (a^2 + b^2 / c^2)
Pythagoras shows a^2 + b^2 = c^2
Therefore = 1
Using ratios, (sin(x))^2 + (cos(x))^ 2 == 1

Question 7

1.2.2 What is the trigonometrical identity derived from a ratio?

Answer 7

sin(x)/cos(x) == tan(x)

This is useful for simplifying equations, and rearrangement of equations.

Proof:

(o/h)/(a/h) = (oh/ah) = (o/a)
therefore sin/cos == tan

Question 8

1.3.1 What is the ‘principal angle’?

Why might this be a problem?

Answer 8

When using calculators to computing trig functions, the vast majority of calculators will only provide one answer. This answer is known as the ‘principal angle’.

Looking at the trig graphs, we can see that there are technically infinitely many values for the computed angle. A question that will crop up on exam papers is to solve a trig equation within a certain range of degrees or radians.

Question 9

1.3.2 What’s the first step to finding the solution of trig equations?

Answer 9

Observe the equation and notice that the functions do not match. In order to solve the equation, the functions must match but may be to a higher order.

E.g.:

(sin(x))^2 + 5cos(x) = 3

Rewritten using identities:

(1-cos(x))^2 + 5cos(x) = 3

Question 10

1.3.3 What is the following step?

Answer 10

Setting the equation equal to zero.

(1-cos(x))^2 + 5cos(x) = 3

-cos(x)^2 + 5cos(x) - 2 = 0

By now, it resembles a quadratic equation, which is exactly how you’re to solve for the cos(x) component. If it’s easier, the negative coefficient of the square (if it exists in the equation) can be eliminated.

cos(x)^2 - 5cos(x) + 2 = 0

Question 11

1.3.4 Applying the quadratic formula.

Answer 11

When applying the quadratic formula, care that it is solving the cos(x), not x itself. That step comes after computing the function.

If possible, keep the roots of the equation in surd form for accuracy.

Question 12

1.3.5 Solving for x.

Answer 12

Using the inverse function on the calculator, you will find the ‘principle angle’ of the equation. This is one answer, however in exams there will be a set range of degrees or radians for the equation to be solved in.

Using prior knowledge of the graphs, you should be able to evaluate the other angles from the equation.

Question 13

1.3.6 How to write the answers.

Answer 13

The answers should be writing as x = a, x= b … etc.

This is to avoid confusion and make the answers clearer.