Reduction to the first quadrant and the 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:

• $\sin 330º$

• $\cos 315º$

• $\tan 300º$

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.

Cian is solving an exercise where he knows that $\sen \alpha = 0.6$ and that angle $\alpha$ belongs to the second quadrant ($90º < \alpha < 180º$). In his notebook he has written the following:

"By the fundamental identity, $\cos \alpha = \sqrt{1 - \sen^2 \alpha} = \sqrt{1 - 0.6^2} = \sqrt{0.64} = 0.8$. Therefore, applying the formula for supplementary angles, $\cos(180º - \alpha) = -0.8$".

1. a) Identify the two conceptual errors that Cian has made and explain, using the meaning of the coordinates on the unit circle, why they are wrong.
2. b) Correct the procedure and calculate the correct values of $\cos \alpha$ and $\cos(180º - \alpha)$.

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.)

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.

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.

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.

• 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.

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 anticlockwise, the height of the tip relative to the centre 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 centre. What values of $\theta$ between $0º$ and $360º$ correspond to this height?

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

In a wind farm in Albacete, a wind turbine blade of length $40\text{ m}$ rotates counterclockwise around its central hub. Taking the horizontal axis passing through the hub as reference ($0º$ to the right), the vertical height of the blade tip relative to the hub is $h(\alpha) = 40 \times \sen \alpha$ (in meters) and its horizontal position is $d(\alpha) = 40 \times \cos \alpha$ (in meters).

During a technical inspection, the blade comes to a stop at a position $\alpha$ such that its horizontal displacement is $d = -24\text{ m}$ and the blade tip is below the hub ($h < 0$).

1. a) Deduce in which quadrant the blade is located and calculate the exact value of $\cos \alpha$.
2. b) Using the fundamental trigonometric identity, calculate the exact height $h$ of the blade tip relative to the hub, rigorously justifying the sign of the root.
3. c) The maintenance sensors only process acute angles $\beta \in [0º, 90º]$. Express $\alpha$ in the form $\alpha = 180º + \beta$ and determine the exact value of $\sen \beta$, $\cos \beta$, and $\tg \beta$. What does $\beta$ represent geometrically with respect to the blade's position?

To calculate the exact value of $\sen(180º + \alpha)$ and $\cos(360º - \alpha)$ knowing that $\tg \alpha = -2$ and that $\alpha$ is in the second quadrant, two classmates propose different strategies:

• Aoife proposes using the identity $1 + \tg^2 \alpha = \frac{1}{\cos^2 \alpha}$ to obtain $\cos \alpha$, then deducing $\sen \alpha$ with $\sen \alpha = \tg \alpha \times \cos \alpha$ and, finally, reducing the requested angles to the first quadrant.

• Cian proposes setting up a system of two equations with two unknowns: $\frac{\sen \alpha}{\cos \alpha} = -2$ and $\sen^2 \alpha + \cos^2 \alpha = 1$, solving for $\sen \alpha = -2\cos \alpha$, substituting into the fundamental identity, and concluding with the reductions.

1. a) Analyze both methods: are they both correct? Which one is more direct and carries less risk of making a mistake when choosing signs?
2. b) Apply the method you consider most efficient to give the exact, simplified value (with radicals) of $\sen(180º + \alpha)$ and $\cos(360º - \alpha)$.

A drone's onboard computer records an orientation angle $\theta \in [0º, 360º)$ that cannot be read directly due to a screen failure, but the memory retains two reliable pieces of data:

1. $\sen(180º - \theta) = -\frac{\sqrt{3}}{2}$.

2. $\cos(-\theta) < 0$.

1. a) Deduce, with reasoning, which quadrant the angle $\theta$ belongs to.
2. b) Determine the exact value of $\theta$ in degrees within $[0º, 360º)$.
3. c) Write a simple trigonometric equation of the form $\cos x = k$ whose only two solutions in the interval $[0º, 360º)$ are the angle $\theta$ you found and its opposite angle (symmetric with respect to the horizontal axis).

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 $\cos(180º + \alpha)$.