Trigonometry with Obtuse Angles
Learn trigonometry with obtuse angles for NSW Year 12 Mathematics Standard 2 (NESA outcome MST-12-S2-04). An obtuse angle lies between \(90^\circ\) and \(180^\circ\): its sine is positive while its cosine and tangent are negative, and it shares its sine with the related acute angle \(180^\circ-\theta\).
You will use \(\sin\theta=\sin(180^\circ-\theta)\) and \(\cos\theta=-\cos(180^\circ-\theta)\) to find an obtuse angle from its ratio, evaluate ratios such as \(\cos 150^\circ\) on a calculator, and get the sign right — the essential foundation for the sine and cosine rules with non-right-angled triangles.
Theory
An obtuse angle is between \(90^\circ\) and \(180^\circ\). This Year 12 Standard 2 (NSW) guide shows why its sine is positive but its cosine and tangent are negative, how to use \(\sin\theta=\sin(180^\circ-\theta)\) to find an obtuse angle from its ratio, and how to evaluate ratios such as \(\cos 150^\circ\) on a calculator — the foundation for the sine and cosine rules with non-right-angled triangles.
An obtuse angle measures between \(90^\circ\) and \(180^\circ\). For NSW Year 12 Mathematics Standard 2 you need the trigonometric ratios of such angles so you can later apply the sine and cosine rules to non-right-angled triangles.
Think of the angle in the second quadrant of a unit circle. The point on the circle has a positive height (\(\sin\theta>0\)) but a negative horizontal position (\(\cos\theta<0\)); the tangent, being their ratio, is also negative.
Every obtuse angle shares its sine with an acute angle, its related acute angle \(180^\circ-\theta\). This is why \(\sin\theta=\sin(180^\circ-\theta)\): the acute and obtuse angles are supplementary.
For an obtuse angle \(\theta\) (with \(90^\circ<\theta<180^\circ\)):
Calculator values (correct to four decimal places) show the pattern: each obtuse angle has the same sine as its related acute angle, while its cosine and tangent have the same size but the opposite sign.
| \(\theta\) | \(\sin\theta\) | \(\cos\theta\) | \(\tan\theta\) |
|---|---|---|---|
| \(70^\circ\) | \(0.9397\) | \(0.3420\) | \(2.7475\) |
| \(110^\circ\) | \(0.9397\) | \(-0.3420\) | \(-2.7475\) |
| \(50^\circ\) | \(0.7660\) | \(0.6428\) | \(1.1918\) |
| \(130^\circ\) | \(0.7660\) | \(-0.6428\) | \(-1.1918\) |
Finding an obtuse angle from a ratio
- Check the sign. An obtuse angle has \(\sin\theta>0\), \(\cos\theta<0\), \(\tan\theta<0\).
- Related acute angle. Enter the positive value into \(\sin^{-1}\), \(\cos^{-1}\) or \(\tan^{-1}\) to get the acute angle.
- Take the supplement. Subtract the acute angle from \(180^\circ\) to get the obtuse angle \(\theta\).
- State the answer to the required accuracy (nearest degree or nearest minute).
Evaluating a ratio of an obtuse angle
Set the calculator to degree mode and enter the ratio directly, for example \(\cos 150^\circ\). Round only at the end, to the accuracy the question asks for, and check the sign: only the sine of an obtuse angle is positive.
Sine is positive for an obtuse angle, so take the supplement of the related acute angle.
| \(\theta_{\text{acute}}\) | \(=\) | \(\sin^{-1}(0.62) = 38.32^\circ\) |
| \(\theta\) | \(=\) | \(180^\circ - 38.32^\circ\) |
| \(\) | \(=\) | \(141.68^\circ\) |
| \(\theta\) | \(\approx\) | \(142^\circ\) |
Cosine is negative for an obtuse angle; use the positive value for the acute angle.
| \(\theta_{\text{acute}}\) | \(=\) | \(\cos^{-1}(0.35) = 69.51^\circ\) |
| \(\theta\) | \(=\) | \(180^\circ - 69.51^\circ\) |
| \(\) | \(=\) | \(110.49^\circ\) |
| \(\theta\) | \(\approx\) | \(110^\circ\) |
Use the sine rule for the angle. The calculator gives the acute angle, so take its supplement because \(A\) is obtuse.
| \(\dfrac{\sin A}{12}\) | \(=\) | \(\dfrac{\sin 25^\circ}{7}\) |
| \(\sin A\) | \(=\) | \(\dfrac{12\sin 25^\circ}{7} = 0.7245\) |
| \(A_{\text{acute}}\) | \(=\) | \(\sin^{-1}(0.7245) = 46.43^\circ\) |
| \(A\) | \(=\) | \(180^\circ - 46.43^\circ = 133.57^\circ\) |
| \(A\) | \(\approx\) | \(134^\circ\) |
Because \(130^\circ\) is obtuse, \(\cos 130^\circ\) is negative, so the last term is added.
| \(\) | \(=\) | \(36 + 81 - 108\times(-0.6428)\) |
| \(\) | \(=\) | \(117 + 69.42\) |
| \(\) | \(=\) | \(186.42\) |
| \(\) | \(\approx\) | \(186.4\) |
Common pitfalls
Frequently asked questions
How do you find an obtuse angle from its sine?
Enter the value into sin inverse to get the related acute angle, then subtract it from 180 degrees. For example, if sin theta equals 0.62 the acute angle is about 38 degrees, so the obtuse angle is 180 minus 38, which is about 142 degrees.
Is the cosine of an obtuse angle positive or negative?
Negative. An obtuse angle sits in the second quadrant, where the horizontal position on the unit circle is negative, so its cosine is negative. Its tangent is also negative, while its sine stays positive.
What does sin theta equals sin of 180 minus theta mean?
It means an obtuse angle and its acute supplement have exactly the same sine. So sin 130 degrees equals sin 50 degrees. This is why solving sin theta equals a positive value gives two answers between 0 and 180 degrees.
How do you find cos 150 degrees on a calculator?
Check the calculator is in degree mode, then enter cos 150. It gives minus 0.8660 correct to four decimal places. The minus sign is right: 150 degrees is obtuse, so its cosine is negative, and it has the same size as cos 30 degrees, its related acute angle.
Why do obtuse angles matter in Standard 2 trigonometry?
The sine rule can produce an obtuse angle when you solve for an unknown angle, and the cosine rule uses the cosine of an angle that may be obtuse. Getting the sign of the ratio right is essential for non-right-angled triangle problems.
How is an obtuse angle different from a reflex angle?
An obtuse angle is between 90 and 180 degrees. A reflex angle is between 180 and 360 degrees. In Standard 2 trigonometry of triangles the largest angle is always less than 180 degrees, so you only deal with acute, right and obtuse angles.