11.05 Trigonometry of obtuse angles

 

Types of angles

Remember the following classifications of angles:

An acute angle has an angle measure between $0^\circ$0° and $90^\circ$90°

An obtuse angle has an angle measure between $90^\circ$90° and $180^\circ$180°

A reflex angle has an angle measure between $180^\circ$180° and $360^\circ$360°

 

Practice questions

Question 1

Question 2

 

The unit circle

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$1 unit, with no other known measurements:

The two smaller, unknown side-lengths defined by $x$x and $y$y can be calculated:

$\cos\theta=\frac{x}{1}$cosθ=x1​         $\sin\theta=\frac{y}{1}$sinθ=y1​

Since dividing any number by 1 gives a same number answer, $x=\cos\theta$x=cosθ and $y=\sin\theta$y=sinθ. Now consider a Cartesian plane that has a circle drawn with a radius of $1$1, centred at the origin $\left(0,0\right)$(0,0):

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

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

The point on the unit circle is now in quadrant $2$2. Angles are still measured from the positive $x$x-axis, so in the $2$2nd quadrant, angles have measures between $90^\circ$90° and $180^\circ$180°, hence they are obtuse angles. Remember that $x$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 $2$2nd quadrant, $x$x values are negative and $y$y values are positive.

So because $x=\cos\theta$x=cosθ and $y=\sin\theta$y=sinθ, 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^\circ$360°, 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^\circ$90° and $180^\circ$180°.

 

Practice questions

Question 3

Question 4

 

Calculating sine & cosine ratios of obtuse angles

A scientific calculator can evaluate the sine, cosine or tangent of any angle of any magnitude. For example, entering $\sin120^\circ$sin120° in the calculator will generate an answer of $0.8660$0.8660, correct to $4$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 $\alpha$α. Since $y=\sin\theta$y=sinθ, this means that for this obtuse angle, $\sin\theta=\sin\alpha$sinθ=sinα. Because $\theta+\alpha=180^\circ$θ+α=180° (definition of supplementary angles), the following relationship exists for the sine of all obtuse angles:

$\sin\theta=\sin\left(180^\circ-\theta\right)$sinθ=sin(180°−θ)
 

The x-coordinate of this point in the 2nd quadrant is opposite to the x-coordinate of this other point in the 1st quadrant with supplementary acute angle $\alpha$α. Since $x=\cos\theta$x=cosθ, this means that for this obtuse angle, $\cos\theta=-\cos\alpha$cosθ=−cosα. Because $\theta+\alpha=180^\circ$θ+α=180°, the following relationship exists for the cosine of all obtuse angles:

$\cos\theta=-\cos\left(180^\circ-\theta\right)$cosθ=−cos(180°−θ)

 

Practice questions

Question 5

Question 6

 

Two angles for one given sine ratio

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^\circ$0° and $90^\circ$90°.

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^\circ$0° and $180^\circ$180°, 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^\circ$0° and $180^\circ$180° 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.

 

Worked examples

Example 1

Solve $\sin\theta=0.5$sinθ=0.5, given that $\theta$θ can be any angle between $0^\circ$0° and $180^\circ$180°.

Think: Using the scientific calculator only, $\theta=\sin^{-1}(0.5)=30^\circ$θ=sin−1(0.5)=30°. But the angle is between $0^\circ$0° and $180^\circ$180°. So since obtuse angles also produce a positive sine ratio, there must be an obtuse angle that would also be a solution to the above equation.

Do: Using the relationship from before, $\sin\theta=\sin\left(180^\circ-\theta\right)$sinθ=sin(180°−θ), this means that $\sin30^\circ=\sin(180^\circ-30^\circ)=\sin150^\circ$sin30°=sin(180°−30°)=sin150°

Therefore $\theta$θ has two possible solutions, $\theta=30^\circ$θ=30° or $\theta=150^\circ$θ=150°. Both solve the equation given above. The unit circle diagram below shows these two solutions for $y=\sin\theta$y=sinθ:

Example 2

Solve $\cos\theta=0.7071$cosθ=0.7071, correct to $1$1 decimal place, given that $\theta$θ can be any angle between $0^\circ$0° and $180^\circ$180°.

Think: Using the scientific calculator only, $\theta=\cos^{-1}(0.7071)=45.0^\circ$θ=cos−1(0.7071)=45.0°. Are there any obtuse angle solutions that could also create a positive cosine ratio? No, because remember, the cosine of an obtuse angle always creates a negative ratio. Since the above ratio is positive, there will only be the one acute angle solution to consider between $0^\circ$0° and $180^\circ$180°.

Therefore, $\theta$θ has only one possible solution, $\theta=45.0^\circ$θ=45.0° is the only angle between $0^\circ$0° and $180^\circ$180° that solves the above equation. The unit circle diagram below shows this one solution for $x=\cos\theta$x=cosθ:

 

Practice questions

Question 7

Question 8