In earlier chapters it was shown how trigonometric functions can be defined for angles of any size, positive or negative, by means of the unit circle definitions. It was also shown how angles outside of the range $0^{\circ}$0∘ to $90^{\circ}$90∘ can be considered to be related to angles within that range.
In the diagram, the point on the unit circle that makes the angle $\theta$θ with the positive $x$x-axis is said to have coordinates $\left(\cos\theta,\sin\theta\right)$(cosθ,sinθ).
In the diagram below, the point that makes the angle $\alpha$α with the positive $x$x-axis has coordinates $\left(\cos\alpha,\sin\alpha\right)$(cosα,sinα).
It is apparent, on comparing the two diagrams, that $\sin\theta=\sin\alpha$sinθ=sinα and $\cos\theta=-\cos\alpha$cosθ=−cosα.
Moreover, using a similar triangles argument we can see that $\theta=180^{\circ}-\alpha$θ=180∘−α or, in radian measure, $\theta=\pi-\alpha$θ=π−α.
Thus, to evaluate a trigonometric function of an obtuse angle in the unlikely situation of having no calculator but only a table of sines and cosines, we could first subtract the obtuse angle from $180^{\circ}$180∘ to find the relative acute angle, and then look up the relevant table, bearing in mind that the cosine of the obtuse angle is the negative of the cosine of the relative acute angle.
To obtain the tangent of an obtuse angle we could first find the sine and cosine of the relative acute angle, then reverse the sign of the cosine, and finally, make use of the definition $\tan x=\frac{\sin x}{\cos x}$tanx=sinxcosx.
More simply, we could look up the tangent of the relative acute angle and make it negative, since the tangent of an obtuse angle must be negative according to the definition.
More simply still, we could just use a calculator directly on the obtuse angle. But it is useful to know how these things are worked out. Also, an understanding of this idea is needed when we are solving trigonometric equations for all solutions between $0^{\circ}$0∘ and $360^{\circ}$360∘.
We extend the same kind of reasoning for angles beyond $180^{\circ}$180∘. In every case, there is a relative acute angle and the coordinates of the point on the unit circle that generates the angle show whether the sign should be positive or negative.
It is useful to remember that for an angle in the second quadrant, $90^{\circ}$90∘ to $180^{\circ}$180∘, the sine function is positive while the others are negative; for an angle in the third quadrant, $180^{\circ}$180∘ to $270^{\circ}$270∘, tangent is positive while the others are negative; and for an angle in the fourth quadrant, $270^{\circ}$270∘ to $360^{\circ}$360∘, cosine is positive and the others are negative.
Examples
Example 1
Evaluate $\tan135^{\circ}$tan135∘.
We find immediately, from a calculator, that the answer should be $-1$−1. We can check that this is correct by first finding the relative acute angle. This is $180^{\circ}-135^{\circ}=45^{\circ}$180∘−135∘=45∘. We know from considering a right-angled isosceles triangle, that $\sin45^{\circ}=\frac{1}{\sqrt{2}}$sin45∘=1√2 and $\cos45^{\circ}=\frac{1}{\sqrt{2}}$cos45∘=1√2. Therefore, $\sin135^{\circ}=\frac{1}{\sqrt{2}}$sin135∘=1√2 and $\cos135^{\circ}=-\frac{1}{\sqrt{2}}$cos135∘=−1√2. Hence, $\tan135^{\circ}=\frac{\sin135^{\circ}}{\cos135^{\circ}}=\frac{\frac{1}{\sqrt{2}}}{-\frac{1}{\sqrt{2}}}=-1.$tan135∘=sin135∘cos135∘=1√2−1√2=−1.
Example 2
Solve for all values of $x$x between $0^{\circ}$0∘ and $360^{\circ}$360∘, $\cos x=-0.454$cosx=−0.454. We know that cosine is negative for angles in the second and third quadrants. So, we expect to find two solutions.
Each of the solution angles must have the relative acute angle whose cosine is $0.454$0.454. Using the inverse cosine function, we determine that the relative acute angle is very close to $63^{\circ}$63∘. So, the solutions we seek are $180^{\circ}-63^{\circ}=117^{\circ}$180∘−63∘=117∘ and $180^{\circ}+63^{\circ}=243^{\circ}$180∘+63∘=243∘. This should be verified by using a calculator to find the cosines of these two angles.
Example 3
Write the following trigonometric ratio using an acute angle:
$\cos165^\circ$cos165°
Example 4
Evaluate $\cos126^\circ$cos126° correct to two decimal places and make note of the sign of your answer.
Example 5
Consider the equation $\sin\theta=-0.6428$sinθ=−0.6428.
Find the measure in degrees of the acute angle satisfying $\sin\theta=0.6428$sinθ=0.6428. Give your answer to the nearest degree.
Find the measure in degrees of the angles satisfying $\sin\theta=-0.6428$sinθ=−0.6428 for $0^\circ\le\theta\le360^\circ$0°≤θ≤360°. Give your answers to the nearest degree.