Trigonometric ratios of an angle
At the Cliffs of Moher visitor centre, a guide tells you the cliffs are about 120 m high and asks how far out to sea you would have to be to see the top at an angle of elevation of 10°. In Junior Cycle you used sine, cosine and tangent in right-angled triangles. For the Leaving Cert you extend them to any angle using a circle of radius $1$, and you use them constantly to find exact values, solve triangles and model anything that rotates or repeats. Here you review the key idea and add the tools you are missing: radians, the special angles, the reciprocal ratios and careful handling of signs. We come back to the cliffs at the end.
From the sides to the unit circle
In a right-angled triangle with an acute angle $\alpha$, the sine is the ratio of the opposite side to the hypotenuse, and the cosine is the ratio of the adjacent side to the hypotenuse. If that triangle is placed with the vertex of the angle at the origin and the hypotenuse as the radius of a circle of radius $1$, the sine and cosine are, directly, the coordinates of the point where the arm of the angle meets the circle.
Calculate the exact value of the sine, cosine and tangent of the smallest angle of a right-angled triangle, knowing that its shorter sides measure $5$ cm and $12$ cm.
Calculate the exact value of the sine, cosine and tangent of the smallest angle of a right-angled triangle, knowing that its shorter sides measure $5$ cm and $12$ cm.
In a right-angled triangle, the tangent of one of its acute angles is $\frac{3}{4}$. Determine the exact value of the secant and cosecant of this angle.
In a right-angled triangle, the tangent of one of its acute angles is $\frac{3}{4}$. Determine the exact value of the secant and cosecant of this angle.
Radians and the special angles: $30°$, $45°$ and $60°$
There are three angles whose sines, cosines and tangents can be calculated exactly from two very particular triangles: the isosceles right-angled triangle (for $45°$) and half of an equilateral triangle (for $30°$ and $60°$). That is why their values are fractions with surds, not rounded decimals.
Convert the angles of $135^\circ$ and $330^\circ$ to radians, and express the angles of $\frac{5\pi}{6}$ rad and $\frac{7\pi}{4}$ rad in sexagesimal degrees.
Convert the angles of $135^\circ$ and $330^\circ$ to radians, and express the angles of $\frac{5\pi}{6}$ rad and $\frac{7\pi}{4}$ rad in sexagesimal degrees.
Evaluate the following expression by replacing the trigonometric ratios with their exact values, without using a calculator:…
Evaluate the following expression by replacing the trigonometric ratios with their exact values, without using a calculator:
$\sin^2(30^\circ) + \cos^2(45^\circ) - \tan(60^\circ) \times \sin(60^\circ)$
Calculate the numerical value of the following trigonometric expression corresponding to angles located on the semi-axes:…
Calculate the numerical value of the following trigonometric expression corresponding to angles located on the semi-axes:
$3\sin(90^\circ) - 2\cos(180^\circ) + 4\sin(270^\circ) - \cos(360^\circ)$
A grinder disc rotates at $7200$ revolutions per minute. Express this angular velocity in radians per second and calculate the exact angle (in radians) swept out by a…
A grinder disc rotates at $7200$ revolutions per minute. Express this angular velocity in radians per second and calculate the exact angle (in radians) swept out by a particle on its outer edge in $0.15$ seconds.
The Pythagorean identity and the sign in each quadrant
Since $P(\cos\alpha, \sin\alpha)$ is always at distance $1$ from the origin (the radius of the unit circle), Pythagoras directly gives the most widely used relationship in the whole of trigonometry. But when you solve for a ratio using a square root, the sign is not decided by the root: it is decided by the quadrant in which the angle lies.
Knowing that $\cos \alpha = \frac{4}{5}$ and that $\alpha$ is an acute angle, find the exact value of $\sin \alpha$ and $\tan \alpha$ using fundamental trigonometric…
Knowing that $\cos \alpha = \frac{4}{5}$ and that $\alpha$ is an acute angle, find the exact value of $\sin \alpha$ and $\tan \alpha$ using fundamental trigonometric identities.
Determine which quadrant the angle $\beta$ lies in if both $\sin \beta < 0$ and $\tan \beta > 0$ hold simultaneously.
Determine which quadrant the angle $\beta$ lies in if both $\sin \beta < 0$ and $\tan \beta > 0$ hold simultaneously.
Given that $\tan \gamma = -\frac{12}{5}$ and that $\gamma$ belongs to the second quadrant, calculate the exact value of $\sin \gamma$ and $\sec \gamma$.
Given that $\tan \gamma = -\frac{12}{5}$ and that $\gamma$ belongs to the second quadrant, calculate the exact value of $\sin \gamma$ and $\sec \gamma$.
It is known that an angle $\alpha$ belongs to the second quadrant and that $\tan\alpha = -\frac{4}{3}$. To calculate the exact values of $\cos\alpha$ and $\sin\alpha$,…
It is known that an angle $\alpha$ belongs to the second quadrant and that $\tan\alpha = -\frac{4}{3}$. To calculate the exact values of $\cos\alpha$ and $\sin\alpha$, two classmates propose two different strategies:
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Method A (Aoife): Applies the identity $1 + \tan^2\alpha = \frac{1}{\cos^2\alpha}$, solves for $\cos\alpha$ choosing the appropriate sign for the quadrant, and then obtains $\sin\alpha = \tan\alpha \times \cos\alpha$.
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Method B (Seán): Sets up a system of equations with the definition $\frac{\sin\alpha}{\cos\alpha} = -\frac{4}{3}$ and the fundamental Pythagorean identity $\sin^2\alpha + \cos^2\alpha = 1$, expressing one unknown in terms of the other to substitute.
- a) Compare both methods, analyzing which one is more direct, which involves less algebraic computation, and in which one it is easier to make a mistake with the choice of signs.
- b) Apply the method you consider most efficient to find the exact values of $\cos\alpha$ and $\sin\alpha$, justifying the choice of sign at each step based on the quadrant of $\alpha$.
Reducing an angle to the first revolution
To find the sign or the value of the ratios of a large angle, first reduce it to $[0°,360°)$ by subtracting complete revolutions, and then locate its quadrant.
Find the equivalent angle in the first revolution (between $0^\circ$ and $360^\circ$) for the angle of $1950^\circ$ and determine the sign of its secant and cotangent.
Find the equivalent angle in the first revolution (between $0^\circ$ and $360^\circ$) for the angle of $1950^\circ$ and determine the sign of its secant and cotangent.
Reducing to the first quadrant
Once the angle is in $[0°,360°)$, find its reference angle $\theta$ (the acute angle it makes with the horizontal axis). The ratio has the same value as that of $\theta$, with the sign given by the quadrant (CAST).
Find the exact value of the following expression by first reducing all the angles to the first quadrant:…
Find the exact value of the following expression by first reducing all the angles to the first quadrant:
$\cos(120^\circ) \times \sin(225^\circ) - \tan(330^\circ) \times \cos(210^\circ)$
Simplify the following trigonometric expression as much as possible, assuming that $x$ is an angle in the first quadrant. Pay special attention to the signs when…
Simplify the following trigonometric expression as much as possible, assuming that $x$ is an angle in the first quadrant. Pay special attention to the signs when reducing to that quadrant:
$\frac{\sin(\pi - x) \times \cos\left(\frac{\pi}{2} + x\right) \times \tan(2\pi - x)}{\cos(\pi + x) \times \sin\left(\frac{3\pi}{2} - x\right)}$
Applications: slope, angles of elevation and simplifying expressions
Trigonometric ratios describe any inclination (the slope of a line is $m=\tan\theta$): the height of a building from its shadow, the tilt of a solar panel, or an algebraic expression that is simplified using the Pythagorean identity.
Back to the Cliffs of Moher: with $\tan 10°\approx0.176$, the cliff of height $120$ m is seen at $10°$ from a distance of $120\div\tan10°\approx680$ m.
A building casts a shadow of $24$ metres when the angle of elevation of the sun above the horizon is $53^\circ$. Write the trigonometric expression to calculate the…
A building casts a shadow of $24$ metres when the angle of elevation of the sun above the horizon is $53^\circ$. Write the trigonometric expression to calculate the height of the building and determine its value. If the angle of elevation is halved, does the length of the shadow double? Justify your answer based on the non-linearity of the tangent ratio.
Simplify the following trigonometric expression as much as possible using fundamental identities, expressing the result in terms of a single trigonometric ratio:…
Simplify the following trigonometric expression as much as possible using fundamental identities, expressing the result in terms of a single trigonometric ratio:
$\frac{\sin^3 x + \sin x \times \cos^2 x}{\cos x}$
Prove the following trigonometric identity, showing the algebraic steps and fundamental relations used. Assume that $\alpha$ is an angle for which all ratios are…
Prove the following trigonometric identity, showing the algebraic steps and fundamental relations used. Assume that $\alpha$ is an angle for which all ratios are defined:
$\frac{1 - \sin \alpha}{\cos \alpha} + \frac{\cos \alpha}{1 - \sin \alpha} = 2\sec \alpha$