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.

Marcos 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 Marcos 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 $\cos(180º - \alpha)$.

Consider the angle $\beta = 200º$.

1. a) Before operating, estimate qualitatively: in which quadrant is $\beta$ located? What signs should $\sen 200º$ and $\cos 200º$ have? If you compare $200º$ with $180º$, which of the two ratios (sine or cosine) do you expect to have a greater absolute value?
2. b) Knowing that $200º = 180º + 20º$ and taking as approximate data $\sen 20º \approx 0,3420$ and $\cos 20º \approx 0,9397$, calculate the exact approximate values of $\sen 200º$, $\cos 200º$ and $\tg 200º$ using the reduction formulas to the first quadrant.
3. c) Check whether the results obtained confirm your estimation from section a) and explain why the tangent turns out to be positive even though both the sine and cosine are negative.

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 $\sin \alpha = 2\cdot\cos \alpha$ 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.

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:

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

• Sergio 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 $\cos(180º + \alpha)$.

• 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 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 $h(\alpha) = 40 \cdot \sen \alpha$ (in meters) and its horizontal position is $d(\alpha) = 40 \cdot \cos \alpha$ (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?

Two angles $\alpha$ and $\beta$ in the interval $[0º, 360º)$ belong to different quadrants and simultaneously satisfy the following system of conditions:

1. $\sen(180º - \alpha) + \sen(180º + \beta) = 0$.

2. $\cos(360º - \alpha) - \cos(180º - \beta) = 0$.

1. a) Apply the reduction formulas to the first quadrant in both equations to obtain a direct relation between the trigonometric ratios of $\alpha$ and those of $\beta$.
2. b) From the previous result, deduce the exact geometric relationship linking the angles $\alpha$ and $\beta$ (are they supplementary, opposite, or do they differ by $180º$?). Justify your conclusion by analyzing the signs on the unit circle.
3. c) Knowing additionally that $\sen \alpha = \frac{\sqrt{5}}{3}$ and that $\alpha$ is in the second quadrant, find the exact value of $\cos \beta$ and $\tg \beta$ using the fundamental identity, reasoning the sign of each step without using a calculator.