11.05 Trigonometry of obtuse angles

Remember the following classifications of angles:

• An acute angle has an angle measure between 0\degree and 90\degree

• An obtuse angle has an angle measure between 90\degree and 180\degree

• A reflex angle has an angle measure between 180\degree and 360\degree

Examples

Example 1

Is this angle acute, obtuse, or reflex?

Worked Solution

Create a strategy

Check the degree of rotation of the angle.

Apply the idea

The given angle is an acute angle as it measures less than 90\degree and forms a narrow, small angle.

Example 2

Is this angle acute, obtuse, or reflex?

Worked Solution

Create a strategy

Check the degree of rotation of the angle.

Apply the idea

The given angle is an obtuse angle as it measures between 90\degree and 180\degree, and forms a wide, large angle.

Trigonometry is not limited to right-angled triangles, which by definition always have one right-angle and two acute angles. In fact, trigonometric ratios can be defined for an angle of any magnitude, including obtuse and reflex angles. This concept can be visualised using the unit circle.

Consider a right-angled triangle with a hypotenuse of length 1 unit, with no other known measurements:

The two smaller, unknown side-lengths defined by x and y can be calculated: \cos\theta =\dfrac{x}{1} \qquad \sin \theta=\dfrac{y}{1}

Since dividing any number by 1 gives a same number answer, x=\cos \theta and y=\sin\theta.

Now consider a Cartesian plane that has a circle drawn with a radius of 1, centred at the origin (0,\,0):

This is known as the unit circle, because it is a circle with a radius of 1 unit. Notice how this same right-angled triangle from before now appears inside the unit circle in quadrant 1 (where x and y are both positive), and the hypotenuse of the triangle is now the radius of the circle. So for any angle \theta between 0\degree and 90\degree, the corresponding x and y coordinates on the circle can be found by evaluating trigonometric ratios \cos\theta and \sin\theta to create the coordinate pair \left(\cos\theta,\,\sin\theta\right).

The angle \theta inside the unit circle is always measured starting from the positive x-axis. So what happens when this angle is extended beyond 90\degree?

The point on the unit circle is now in quadrant 2. Angles are still measured from the positive x-axis, so in the 2nd quadrant, angles have measures between 90\degree and 180\degree, hence they are obtuse angles. Remember that x values on the cartesian plane are negative when looking to the left of the origin (centre) and positive to the right of the origin. So in this 2nd quadrant, x values are negative and y values are positive.

So because x=\cos \theta and y=\sin \theta, that means the cosine of an obtuse angle must be a negative trigonometric ratio. And the sine of an obtuse angle is a positive trigonometric ratio.

The unit circle can be used to find the trigonometric ratio of a reflex angle, angles greater than 360\degree, and negative angles too. But in this course, only acute and obtuse angles will be used when solving trigonometry problems.

The unit circle can also be used to evaluate tangent ratios for any positive (or negative) angles. But in this course, only sine and cosine ratios will be evaluated for angles between 90\degree and 180\degree.

Examples

A scientific calculator can evaluate the sine, cosine or tangent of any angle of any magnitude. For example, entering \sin 120\degree in the calculator will generate an answer of 0.8660, correct to 4 decimal places. However, the unit circle can also be used to evaluate trigonometric ratios of obtuse angles, in terms of corresponding acute angles.

Consider an obtuse angle theta. Remember that this angle corresponds to a coordinate pair on the unit circle that is in the 2nd quadrant.

The y-coordinate of this point in the 2nd quadrant is identical to the y-coordinate of this other point in the 1st quadrant with supplementary acute angle \alpha. Since y=\sin \theta, this means that for this obtuse angle, \sin \theta = \sin\alpha. Because \theta + \alpha = 180\degree (definition of supplementary angles), the following relationship exists for the sine of all obtuse angles: \sin \theta =\sin \left(180\degree -\theta \right)

The x-coordinate of this point in the 2nd quadrant is identical to the x-coordinate of this other point in the 1st quadrant with supplementary acute angle \alpha. Since x=\cos \theta, this means that for this obtuse angle, \cos \theta = -\cos\alpha. Because \theta + \alpha = 180\degree, the following relationship exists for the sine of all obtuse angles: \cos \theta =-\cos \left(180\degree -\theta \right)

Examples

Example 4

Evaluate \cos 126\degree correct to two decimal places and make note of the sign of your answer.

Worked Solution

Create a strategy

Determine in which quadrant the angle lies and the sign of the ratio.

Apply the idea

Starting at the positive x-axis and rotating 126\degree anticlockwise, we move into the second quadrant.

This diagram shows cosine is negative in the second quadrant and we know that \cos \theta\degree=-\cos\left(180\degree - \theta\degree\right).

\displaystyle \cos 126\degree	\displaystyle =	\displaystyle -\cos (180\degree -126\degree)	Substitute the angle
	\displaystyle =	\displaystyle -\cos 54\degree	Evaluate the angle
	\displaystyle =	\displaystyle -0.59	Simplify

Reflect and check

The sign of any trigonometric ratio can be determined from its quadrant, using the mnemonic "All Stations To Central" or ASTC.

Watch question walkthrough

Example 5

Write the following trigonometric ratio using an acute angle: \sin 147\degree

Worked Solution

Create a strategy

Determine in which quadrant the angle lies and the sign of the ratio.

Apply the idea

147\degree is between 90\degree and 180\degree so it is in the second quadrant.

This diagram shows sine is positive in the second quadrant and we know that \sin \theta\degree=+\sin\left(180\degree - \theta\degree\right).

\displaystyle \sin 147\degree	\displaystyle =	\displaystyle -\sin (180\degree -147\degree)	Substitute the angle
	\displaystyle =	\displaystyle -\sin 33\degree	Evaluate the angle

Watch question walkthrough

Example 6

Consider the unit semi-circle diagram shown below.

a

State the values of \sin 60\degree and \cos 60\degree, to four decimal places where appropriate.

Worked Solution

Create a strategy

Use your calculator and make sure it is in degrees mode.

Apply the idea

Using a calculator we get: \sin 60\degree=0.866 \qquad \qquad \cos 60\degree=0.5

b

Using point A and the symmetry of the unit circle, state the coordinates of point B.

Worked Solution

Create a strategy

Use the fact that points A and B are symmetrical with respect to the y-axis.

Apply the idea

Points A and B have same coordinate but different signs in x-coordinate (A in 1st quadrant, B in 2nd). So, B: \, (-0.5,\,0.866)

c

Determine the values of \sin 120\degree and \cos 120\degree, to four decimal places where appropriate.

Worked Solution

Create a strategy

Use your calculator and make sure it is in degrees mode.

Apply the idea

Using a calculator we get: \sin 120\degree=0.866 \qquad \qquad \cos 120\degree=-0.5

Reflect and check

Notice that the values are same in the part (a) due to the formulas, \sin \theta=\sin \left(180\degree-\theta \right) and \cos \theta=-\cos \left(180\degree-\theta \right). These formulas state that the sine and cosine of an angle \theta are equal to the sine and cosine of its supplementary angle \left(180\degree-\theta \right), respectively, with a sign change for cosine.

When working with right-angled triangles, unknown angles are found by finding the inverse sine, cosine or tangent of a given ratio of a triangle's side lengths. This is the case for angles between 0\degree and 90\degree.

From the unit circle work completed above, it was discovered that the cosine of acute angles produce positive ratios, whilst the cosine of obtuse angles produce negative ratios. So when trying to find angles between 0\degree and 180\degree, the angle will be either acute or obtuse depending on whether the given ratio is positive or negative.

When it comes to sine ratios, the results are ambiguous, because the sine of any angle between 0\degree and 180\degree will always produce a positive ratio. So without any known point on a unit circle, if an angle could be either acute or obtuse, then a positive sine ratio creates two possible angle solutions.

Consider any position on the unit circle, (x,\,y) can be found using angle \theta: x=\cos \theta \qquad\qquad y=\sin \theta

The following identities can be used to find sine & cosine ratios for obtuse angles \sin \theta =\sin \left(180\degree -\theta \right) \qquad\qquad \cos \theta =-\cos \left(180\degree -\theta \right)

Examples