chevron icon chevron icon chevron icon chevron icon

Further Trigonometry

In this article we will learn about the concepts of

  • Sine and Cosine of obtuse angles
  • Finding area of triangle using trigonometry 
  • Sine Rule
  • Cosine Rule 

This article is specifically written to meet the requirements of the Secondary 3 Mathematics syllabus.

1. Sine of Obtuse Angles

Further Trigonometry Image 1

\(\displaystyle{\mathrm{sin} \;\theta = \mathrm{sin}(180°-\theta)}\)

We know that,

\(\begin{align*} \mathrm{sin \; \theta} &= \mathrm{\frac{opposite} {hypotenuse}} ​​\\[2ex] \therefore \qquad\quad \mathrm{sin \; \theta} &= \frac{a}{c} \\[2ex] \mathrm{sin(180^\circ - \theta)} &= \mathrm{sin \; \theta} \\ \\ &=\frac{a}{c} \\ \end{align*}\)

2. Cosine of Obtuse Angle

Further Trigonometry Image 2

\(\begin{align} \displaystyle{\mathrm{cos \;\theta} = -\mathrm{cos(180°- \theta)}} \end{align}\)

We know that,

\(\begin{aligned} \mathrm{cos \,\theta} &= \frac {b}{c} \\[2ex] \mathrm{cos(180°-\theta)} &= \mathrm{-cos \,\theta} \\[2ex] &= - \frac {b}{c} \end{aligned}\)

3. Area of Triangle

Further Trigonometry Image3

The above mentioned triangle is not a right angle triangle and hence its area cannot be given by the conventional formula.

\(\displaystyle{\text{Area Of △ABC}} = \displaystyle{\mathrm{\frac {1}{2}} ab \mathrm{\;sin\;C}}\)

4. Sine Rule

Further Trigonometry Image 4

Sine rule gives us

\(\displaystyle{\frac {\mathrm{sin \;A}}{a} = \frac {\mathrm{sin \;B}}{b} = \frac {\mathrm{sin \;C}}{c}} \)

or

\(\displaystyle{\frac {a}{\mathrm{sin \;A}} = \frac {b}{\mathrm{sin \;B}} = \frac {c}{\mathrm{sin \;C}}}\)

The smaller letters signify the length of the side whereas the capital letters indicate the angles.

The side(s) correspond opposite to the angle(s) given, and the question will always have at least one of the sides or angles mentioned in it.

The similar coloured letters are always paired together.

5. Cosine Rule

Case 1:

In this situation, we usually find the length of the side when 2 sides and 1 included angle is given.

Further Trigonometry Image 5

We need to find the value of \(a\) when values of \(b\), \(c\) & \(\mathrm{A}\) are given to us.

You can use the following formula to obtain the value of \(a\)

\(\begin{aligned} a^2 &= b^2 + c^2- 2bc \;\mathrm{cos\,A} \\ \\ \implies\quad a &= \sqrt {b^2 + c^2 - 2bc \; \mathrm{cos\,A}} \end{aligned}\)

 

Case 2:

In this situation, we are expected to find the angle when the values of \(3\) sides are given to us.

Further Trigonometry Image 6

Taking the example here, we are given the values of \(a,b,c\) and the value of angle \(\mathrm{A}\) is to be found. In this case, we use the formula,

\(\begin{aligned} \mathrm{cos\,A} &= \frac {b^2 + c^2 - a^2} {2bc} \\[2ex] \implies\quad \mathrm{A} &= \mathrm{cos^{-1}} \bigg( \frac {b^2 + c^2 - a^2} {2bc} \bigg) \end{aligned}\)

 

Example 1:

The figure shows Quadrilateral \(\mathrm{ABCD}\), where \(\mathrm{AB}\) is parallel to \(\mathrm{DC}\).

Find

  1. the length of \(\mathrm{AC}\)
  2. angle \(\mathrm{ADC}\)

Further Trigonometry Image 7

Solution:

For Part (A):

When we look at the \(\mathrm{\triangle ADC}\) we do not have sufficient information, since we need to know at least three things to go ahead with our calculations.

However upon looking at \(\mathrm{\triangle ABC}\) we see there are three parameters given to us, that is two sides and \(1\) included angle which satisfies the condition for case 1 in Sine Rule and hence, we can use the formula   \(a^2= b^2+ c^2- 2.b.c. \;\mathrm{cos\,A}\) to find out the length of \(\mathrm{AC}\).

Hence,

\(\begin{align*} \mathrm{AC} &= \sqrt {6^2+ 5^2- 2 \times 6 \times 5 \times \mathrm{cos}\,53°} \\[2ex] \mathrm{AC} &= \mathrm{4.989 \;cm ≈ 4.99\;cm} \end{align*}\)

 

For Part (B):

Now that we have the length of 2 sides and an angle you can use the sine rule to determine the value of angle \(ADC\)

\(\begin{align*} \mathrm{\frac {sin \,\angle ADC}{AC}} &= \mathrm{\frac {sin \,\angle DAC}{CD}}\\[2ex] \implies\> \mathrm{\frac {sin \,\angle ADC} {4.989}} &= \mathrm{\frac {sin \,82°}{7}} \\[2ex] \implies\>\>\> \mathrm{sin \,\angle ADC} &= \mathrm{\frac {4.989 \;sin \;82°} {7}} \\[2ex] \implies\qquad \mathrm{\angle ADC} &= \mathrm{sin^{-1} \bigg( \frac {4.989 \;sin \;82°} {7} \bigg)} \\[2ex] \implies\qquad \mathrm{\angle ADC} &= \mathrm{44.893° ≈ 44.9°} \end{align*}\)

Conclusion

In this article, we have discussed formulas relating to Sine & Cosine of Obtuse Angles, and the concepts and techniques on how to find the area of a triangle using trigonometry, the formulae for Sine Rule and Cosine Rule and ways to apply them in various situations
 

Continue Learning
Further Trigonometry Quadratic Equations And Functions
Linear Inequalities Laws of Indices
Coordinate Geometry Graphs Of Functions And Graphical Solution
Applications Of Trigonometry
Resources - Academic Topics
icon expand icon collapse Primary
icon expand icon collapse Secondary
icon expand icon collapse
Book a free product demo
Suitable for primary & secondary
select dropdown icon
Our Education Consultants will get in touch with you to offer your child a complimentary Strength Analysis.
Book a free product demo
Suitable for primary & secondary
icon close
Default Wrong Input
Get instant access to
our educational content
Start practising and learning.
No Error
arrow down arrow down
No Error
*By submitting your phone number, we have
your permission to contact you regarding
Geniebook. See our Privacy Policy.
Success
Let’s get learning!
Download our educational
resources now.
icon close
Error
Error
Oops! Something went wrong.
Let’s refresh the page!
Geniebook CTA Illustration Geniebook CTA Illustration
Turn your child's weaknesses into strengths
Geniebook CTA Illustration Geniebook CTA Illustration
close icon
close icon
Turn your child's weaknesses into strengths
Trusted by over 220,000 students.
 
Arrow Down Arrow Down
 
Error
Oops! Something went wrong.
Let’s refresh the page!
Error
Oops! Something went wrong.
Let’s refresh the page!
We got your request!
A consultant will be contacting you in the next few days to schedule a demo!
*By submitting your phone number, we have your permission to contact you regarding Geniebook. See our Privacy Policy.