Exercises: Trigonometric ratios in right-angled triangles

Look at the right-angled triangle in the figure.

1. a) Write, as fractions, the ratios $\sen\ \alpha$, $\cos\ \alpha$ and $\tg\ \alpha$ of the marked angle. b) Write these same ratios for the other acute angle. What relationship do you observe between them?

In the right-angled triangle in the figure, the legs measure $6$ cm and $8$ cm.

1. a) Calculate the hypotenuse. b) Find the three trigonometric ratios of the marked angle $\beta$.

In a right-angled triangle, the hypotenuse measures $13$ cm and the side opposite angle $\alpha$ measures $5$ cm. a) Calculate the other side. b) Find $\sen\ \alpha$, $\cos\ \alpha$ and $\tg\ \alpha$.

The acute angles of a right-angled triangle are $35^\circ$ and $55^\circ$. Knowing that $\sen\ 35^\circ \approx 0{,}574$, find without a calculator the value of $\cos\ 55^\circ$ and justify it using the definition of trigonometric ratios.

In the triangle in the figure, the hypotenuse measures $10$ cm and angle $\alpha$ is $30^\circ$.

Calculate the length of the side opposite $\alpha$. (Given: $\sen\ 30^\circ = 0{,}5$.)

In a right-angled triangle, the side adjacent to an angle of $60^\circ$ measures $8$ m. Calculate the hypotenuse. (Given: $\cos\ 60^\circ = 0{,}5$.)

The two shorter sides of a right-angled triangle each measure $4$ cm. a) What is the tangent of each of the acute angles? b) What are the measurements of these angles? Justify your answer without a calculator.

Determine whether these statements about an acute angle $\alpha$ are true or false: a) "The sine of $\alpha$ can be greater than $1$." b) "The tangent of $\alpha$ can be greater than $1$." Justify both answers using the definition of the ratios in a right-angled triangle and give an example where applicable.

In a 4th year ESO class, Lucas has to calculate the trigonometric ratios of the angle of inclination $\alpha$ of a ramp that rises $1{,}5\text{ m}$ in height and has a horizontal base on the ground of $4\text{ m}$. Lucas writes in his notebook:

"$\sen\ \alpha = \frac{1{,}5}{4} = 0{,}375$".

1. a) Explain in your own words what conceptual error Lucas made when identifying the sides of the right-angled triangle. What trigonometric ratio did he actually calculate instead of the sine?
2. b) Without calculating the angle with a calculator, reason why the actual value of $\sen\ \alpha$ must necessarily be less than $0{,}375$, based on the size relationship between the hypotenuse and the adjacent side.
3. c) Calculate the exact length of the hypotenuse using the Pythagorean theorem and find the corrected value of $\sen\ \alpha$ rounded to three decimal places.

Spanish accessibility regulations in public buildings (Technical Building Code, CTE) state that short ramps for people in wheelchairs must have a maximum slope of $10\,\%$, which is equivalent to saying that the tangent of the angle of inclination cannot exceed the value $\tg\ \alpha \le 0{,}10$.

At the entrance of a municipal library, there is a step of height $18\text{ cm}$ ($0{,}18\text{ m}$) to overcome, with an available horizontal distance on the sidewalk of $2\text{ metres}$.

1. a) Identify the opposite and adjacent legs of the right-angled triangle modeling the ramp, calculate the tangent of the angle of incline $\alpha$, and determine whether the ramp complies with the CTE regulations.
2. b) A team member proposes installing a standard ramp that forms a special angle of inclination of $30^\circ$, thinking that "since it is an angle of only $30^\circ$, the climb will be very gentle". Consult the table of trigonometric ratios for special angles (take $\sqrt{3} \approx 1{,}732$) to calculate $\tg\ 30^\circ$ and argue why this proposal would represent a serious accessibility violation.

For an acute angle $\alpha$, it is known that $\sen\ \alpha = 0{,}6$. Calculate $\cos\ \alpha$ and $\tg\ \alpha$ using trigonometric identities.

If $\tg\ \alpha = 2$ and $\alpha$ is acute, calculate $\sen\ \alpha$ and $\cos\ \alpha$.

1. a) Look at Panel A in the figure, where an equilateral triangle of side $2$ has been divided by an altitude, and deduce the exact values of $\sen 30^\circ$, $\cos 30^\circ$, and $\tg 30^\circ$.

1. b) Look at Panel B, where a square of side $1$ has been divided by a diagonal, and deduce the trigonometric ratios of $45^\circ$ ($\sen 45^\circ$, $\cos 45^\circ$, and $\tg 45^\circ$).

1. c) Using the above and the relationships between complementary angles, complete the following table with the ratios of $30^\circ$, $45^\circ$, and $60^\circ$:

Knowing that $\cos\ \alpha = \frac{1}{3}$ and that $\alpha$ is acute, find the exact values of $\sen\ \alpha$ and $\tg\ \alpha$, expressed in surd form.

A $5$ m ladder rests against a wall, forming an angle of $65^\circ$ with the ground, as shown in the figure.

1. a) What height does the ladder reach on the wall? b) How far from the wall is its foot?

A kite flies attached to a taut string of $40$ m that forms an angle of $50^\circ$ with the ground, as shown in the figure.

If the hands of the person holding it are $1{,}5$ m off the ground, at what height is the kite flying?

An access ramp covers a vertical rise of $0{,}8$ m with a horizontal distance of $6$ m, as shown in the figure.

1. a) What angle does the ramp form with the ground? b) What is its gradient expressed as a percentage? c) Accessibility regulations recommend that these ramps do not exceed a gradient of $10\ \%$: does this ramp comply with them?

A student writes: "as $\sen\ \alpha = 0{,}9$, then $\cos\ \alpha = 1 - 0{,}9 = 0{,}1$". Explain what relationship they confused, correct the mistake and calculate the correct value of $\cos\ \alpha$.

In an industrial design workshop, a metal support part shaped like a right-angled triangle needs to be manufactured. The side lengths are not known, but the technical specifications require that the cosine of the acute angle $\alpha$ be exactly four times its sine, that is, $\cos\ \alpha = 4 \cdot \sen\ \alpha$.

1. a) From the definition $\tg\ \alpha = \frac{\sen\ \alpha}{\cos\ \alpha}$, deduce the exact value of the tangent of $\alpha$ as an irreducible fraction.
2. b) Knowing that the tangent is the quotient of the opposite leg divided by the adjacent leg, deduce which leg is the shorter one. If the final design requires that this shorter leg measures exactly $6\text{ cm}$, determine how long the other leg must be and calculate the length of the hypotenuse (rounded to two decimal places).
3. c) Using the side lengths you have determined, calculate the values of $\sen\ \alpha$ and $\cos\ \alpha$, and verify numerically that the fundamental trigonometric identity $\sen^2\alpha + \cos^2\alpha = 1$ holds.

On Spain's State road network, a DGT hazard warning sign indicates a steep descent with a gradient of $10\,\%$, which technically means that $\tg\ \alpha = 0{,}10$. A vehicle travels along that section for exactly $1\,000\text{ metres}$ measured along the asphalt (that is, along the hypotenuse of the right-angled triangle).

1. a) ESTIMATE: Before performing any trigonometric calculation, estimate at a glance whether the vertical drop the car descends will be closer to $10\text{ m}$, $100\text{ m}$, or $500\text{ m}$. Explain intuitively why you discard the other two options based on what a $10\,\%$ inclination means.
2. b) CHECK: To calculate the actual vertical drop ($h$), you need the opposite leg from the hypotenuse, which requires knowing $\sen\ \alpha$. Knowing that for this angle the cosine is approximately $\cos\ \alpha \approx 0{,}995$, use the relationship $\tg\ \alpha = \frac{\sen\ \alpha}{\cos\ \alpha}$ to calculate $\sen\ \alpha$ and find the exact metres of vertical drop the car descends.
3. c) Compare your result with the $100\text{ metres}$ that a quick calculation estimating $10\,\%$ directly on the distance travelled would yield. Is it reasonable in everyday driving to confuse the distance along the asphalt with the horizontal distance? Reason your answer based on the error obtained.

1. a) Expand $(\sen\ \alpha + \cos\ \alpha)^2$ and use the fundamental trigonometric identity to prove that $(\sen\ \alpha + \cos\ \alpha)^2 = 1 + 2 \cdot \sen\ \alpha \cdot \cos\ \alpha$. b) If for a certain acute angle $\sen\ \alpha + \cos\ \alpha = 1{,}3$ holds true, calculate the value of $\sen\ \alpha \cdot \cos\ \alpha$ without finding the angle.

Knowing that $\tg\ \alpha = \frac{3}{2}$, calculate the value of $\frac{\sen\ \alpha + \cos\ \alpha}{\sen\ \alpha - \cos\ \alpha}$ without calculating the angle. (Hint: divide both numerator and denominator by $\cos\ \alpha$.)

On a mountain road there is a sign warning of a gradient of $12\ \%$, as shown in the figure.

1. a) Explain what this $12\ \%$ means in terms of vertical change and horizontal distance. b) What angle does the road form with the horizontal? c) If a cyclist covers $250$ m along this road, measured along the tarmac, what vertical height have they climbed?

A triangle has two sides of $8$ cm and $12$ cm forming an angle of $50^\circ$. a) Calculate the height perpendicular to the side of $12$ cm using the sine of $50^\circ$. b) Calculate the area of the triangle. c) Generalise: write a formula for the area of a triangle given two sides $a$ and $b$ and the included angle $\gamma$.

A pole of $3{,}2$ m casts a shadow of $4{,}5$ m on the ground, as shown in the figure.

1. a) What is the angle of elevation of the Sun at that moment? b) At that same time, what will be the length of the shadow of a $18$ m building?

In a right-angled triangle, the side adjacent to angle $\beta$ measures $8$ cm and $\cos\ \beta = 0{,}8$. a) Calculate the hypotenuse. b) Calculate the perimetre of the triangle. c) Calculate its area.

Let a right-angled triangle have acute angles $\alpha$ and $\beta$. a) Verify using the triangle with shorter sides measuring $5$ cm and $12$ cm that $\sen^2\ \alpha + \sen^2\ \beta = 1$. b) Prove that this equality always holds, using the fact that $\alpha + \beta = 90^\circ$ and the relationships between the ratios of complementary angles.

Calculate without a calculator, using the table of special angles: a) $\sen^2\ 30^\circ + \cos^2\ 60^\circ + \tg\ 45^\circ$. b) $\frac{\sen\ 60^\circ}{\cos\ 30^\circ} + \tg^2\ 30^\circ$.

To calculate the remaining trigonometric ratios of an acute angle when one of them is known, there are two common procedures:

• Method A (Algebraic): Directly apply the fundamental formulas $\sen^2\alpha + \cos^2\alpha = 1$ and $\tg\ \alpha = \frac{\sen\ \alpha}{\cos\ \alpha}$.

• Method B (Geometric): Interpret the known ratio as the quotient of two sides in an auxiliary right-angled triangle, find the third side using the Pythagorean theorem, and deduce the other ratios.

Two situations are proposed:

• Situation 1: It is known that $\cos\ \alpha = 0{,}6$.

• Situation 2: It is known that $\tg\ \beta = \frac{5}{12}$.

1. a) Solve Situation 1 using Method A to find $\sen\ \alpha$ and $\tg\ \alpha$.
2. b) Solve Situation 2 using Method B to find $\sen\ \beta$ and $\cos\ \beta$.
3. c) CHOOSE THE METHOD: Try setting up Situation 2 using Method A and explain what algebraic complication arises when solving for the unknowns. Justify in a reasoned manner which of the two methods turns out to be much more efficient and reliable when the initial data is the tangent instead of the sine or the cosine.

Following the technical criteria of IDAE (Institute for Energy Diversification and Saving) in Spain, solar panels installed on flat roofs must be separated to prevent one row from casting a shadow on the next at the most critical time of the year: solar noon on the winter solstice.

In an industrial warehouse in Toledo, rows of solar panels are installed whose inclined structure (the hypotenuse of the triangle they form) is $2\text{ metres}$ long. Each panel is mounted with a fixed inclination of $30^\circ$ relative to the ground.

1. a) Using the values of special angles (without a calculator), calculate the exact vertical height reached by the top of the panel above the roof and the horizontal distance occupied by its base on the ground.
2. b) At solar noon on the winter solstice at that location, the sun's rays strike at an elevation angle of $30^\circ$ with the horizontal ground. If the shadow cast backwards by the panel has the panel's vertical height calculated in the previous section as its opposite leg, calculate the length of this shadow on the ground (take $\sqrt{3} \approx 1{,}73\text{ m}$).
3. c) The minimum separation distance between the base of one row and the base of the next must equal the base of the panel plus the length of the projected shadow. If the roof has a total depth of $15\text{ metres}$ from north to south to place panels, justify in a reasoned way the maximum number of complete rows that can be placed without any of them suffering losses due to shadows.