Exercises: Reduction to the first quadrant and fundamental identity

The figure shows the angles $30º$ and $150º$ on the unit circle.

Looking at the symmetry of the figure, express $\sin 150º$ and $\cos 150º$ in terms of $\sin 30º$ and $\cos 30º$, stating the correct sign of each.

Express each of these trigonometric ratios in terms of the ratio of a first-quadrant angle, taking care with the sign:

• $\sin 210º$

• $\cos 225º$

• $\tan 240º$

Express each of these trigonometric ratios in terms of the ratio of a first-quadrant angle, taking care with the sign:

• $\sin 330º$

• $\cos 315º$

• $\tan 300º$

Using a calculator, give the value of these ratios to four decimal places and verify the indicated relationship:

• Calculate $\sin 140º$ and $\sin 40º$. What is the relationship between both values?

• Calculate $\cos 140º$ and $\cos 40º$. What is the relationship between both values?

• Explain, using the unit circle, why what you observed happens.

Given that $\sin \alpha = \frac{3}{5}$ and that $\alpha$ belongs to the second quadrant, calculate the exact value of $\cos \alpha$ and $\tan \alpha$.

Given that $\cos \alpha = -\frac{1}{3}$ and that $\alpha$ belongs to the third quadrant, calculate the exact value of $\sin \alpha$ and $\tan \alpha$.

Given that $\tan \alpha = 2$ and that $\alpha$ belongs to the third quadrant, calculate the exact value of $\sin \alpha$ and $\cos \alpha$. (Hint: use the fact that $\sin \alpha = 2\times\cos \alpha$ and substitute into the fundamental identity.)

The figure shows the unit circle and the horizontal line $\sin x = \frac{1}{2}$, which intersects it at two points.

Find all angles $x$ in $[0º, 360º)$ that satisfy $\sin x = \frac{1}{2}$, reasoning from the figure.

Find all angles $x$ in $[0º, 360º)$ that satisfy $\cos x = -\frac{\sqrt{2}}{2}$, stating which quadrants they belong to.

The figure shows the angles $60º$ and $120º$, which are supplementary.

Given that $\sin 60º = \frac{\sqrt{3}}{2} \approx 0.866$ and $\cos 60º = \frac{1}{2}$, calculate $\sin 120º$ and $\cos 120º$ reasoning from the symmetry of the figure.

Given that $\tan \alpha = -\frac{3}{4}$ and that $\alpha$ belongs to the fourth quadrant:

• Calculate the exact value of $\sin \alpha$ and $\cos \alpha$.

• Calculate $\sin(180º - \alpha)$.

• Calculate $\cos(180º + \alpha)$.

• Find all angles $x$ in $[0º, 360º)$ that satisfy $\tan x = 1$.

• Using the unit circle, reason how many solutions the same equation would have in the interval $[0º, 720º)$ and write them all down.

A student wrote this working:

«If $\cos \alpha = -0.8$ and $\alpha$ is in the second quadrant, then:

$\sin \alpha = \sqrt{1 - 0.64} = \sqrt{0.36} = -0.6$»

Find the mistake made, explain why it is a mistake, and calculate $\sin \alpha$ and $\tan \alpha$ correctly.

Prove that, for any angle $\alpha$, the following holds:

(Hint: first reduce each ratio to the first quadrant.)

Find all angles $x$ in $[0º, 360º)$ that satisfy $\sin^2 x = \frac{1}{4}$, reasoning how many solutions there are and which quadrants they belong to.

The tip of a wind turbine blade describes a circle of radius $1\ \text{m}$ (in relative units) around the rotor axis. If $\theta$ is the angle that the blade makes with the horizontal, measured counterclockwise, the height of the tip relative to the center is $\sin \theta$ and its horizontal displacement is $\cos \theta$.

• Complete a table with the height and horizontal displacement of the tip for $\theta = 0º,\ 90º,\ 180º$ and $270º$.

• The blade rotates $40$ times per minute. After a period of time, the tip is at a height of $0.5\ \text{m}$ above the center. What values of $\theta$ between $0º$ and $360º$ correspond to this height?

• What about a height of $-0.5\ \text{m}$?