Trigonometric Ratios of an Angle
Wilkin wants to know how tall an antenna tower near his school is, but he can't climb it. He measures its shadow on the ground and the angle at which the sun's rays hit the floor. How does he turn those two pieces of data into a height? Trigonometric ratios are the tool, and in this section we extend them to any angle, not just the acute angles of a right triangle. At the end we come back to the tower.
From the legs to the unit circle
In a right triangle with acute angle $\alpha$, the sine is the opposite leg divided by the hypotenuse, the cosine is the adjacent leg divided by the hypotenuse, and the tangent is the opposite leg divided by the adjacent leg. If we place the vertex of the angle at the origin and let the hypotenuse be the radius of a circle of radius $1$, the sine and the cosine become the coordinates of the point where the side 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.
To optimize solar energy collection in winter, a rectangular panel $2$ meters long is raised by resting one of its shorter sides on flat ground, so that the opposite…
To optimize solar energy collection in winter, a rectangular panel $2$ meters long is raised by resting one of its shorter sides on flat ground, so that the opposite side is at a height of exactly $1.2$ meters. Determine the exact value of the cosine and tangent of the angle of inclination of the panel.
Degrees and radians
Angles are measured in degrees or in radians. One radian is the central angle that intercepts an arc whose length equals the radius. Since the whole circumference measures $2\pi r$, one full turn is $2\pi$ radians.
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.
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 special angles: $30^\circ$, $45^\circ$, and $60^\circ$
Three angles have exact ratios, with square roots instead of rounded decimals. They can be justified geometrically with two figures.
Proof with the square. A square of side $1$ has diagonal $\sqrt{1^2 + 1^2} = \sqrt{2}$. The diagonal makes a $45^\circ$ angle with the sides. Then $\sin45^\circ = \dfrac{1}{\sqrt2} = \dfrac{\sqrt2}{2}$, $\cos45^\circ = \dfrac{\sqrt2}{2}$, and $\tan45^\circ = 1$.
Proof with the equilateral triangle. An equilateral triangle of side $2$ is cut by its altitude into two right triangles with hypotenuse $2$, one leg $1$, and the other leg $\sqrt{2^2 - 1^2} = \sqrt3$. The acute angles are $30^\circ$ and $60^\circ$.
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:
$\sin^2(30^\circ) + \cos^2(45^\circ) - \tan(60^\circ) \times \sin(60^\circ)$
The fundamental identity and the sign by quadrant
Since $P(\cos\alpha, \sin\alpha)$ is at distance $1$ from the origin, the Pythagorean theorem gives the most widely used relation in trigonometry. When we solve with a square root, the sign is decided by the quadrant where the angle lies, not by the root.
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 $\sec \gamma$.
Reciprocal ratios and identities
Besides sine, cosine, and tangent there are three more ratios, which are their reciprocals.
How to verify an identity
To prove an identity you work on one side only, usually the more complicated one, until you reach the other. The usual steps are to rewrite everything in terms of $\sin$ and $\cos$, factor out a common factor, add fractions with a common denominator, and use $\sin^2 + \cos^2 = 1$.
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.
Calculate the exact values of $\cot \alpha$ and $\csc \alpha$ given that $\sec \alpha = 3$ and that the sine of the angle is negative.
Calculate the exact values of $\cot \alpha$ and $\csc \alpha$ given that $\sec \alpha = 3$ and that the sine of the angle is negative.
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:
$\frac{1 - \sin \alpha}{\cos \alpha} + \frac{\cos \alpha}{1 - \sin \alpha} = 2\sec \alpha$
Large angles and reduction to the first quadrant
For an angle greater than $360^\circ$, you subtract full turns until it lies between $0^\circ$ and $360^\circ$, and then you locate its quadrant. To get its exact value, you relate it to an acute angle $\theta$ (the reference angle).
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.
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:
$3\sin(90^\circ) - 2\cos(180^\circ) + 4\sin(270^\circ) - \cos(360^\circ)$
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:
$\frac{\sin(\pi - x) \times \cos\left(\frac{\pi}{2} + x\right) \times \tan(2\pi - x)}{\cos(\pi + x) \times \sin\left(\frac{3\pi}{2} - x\right)}$
Equations and relations between ratios
Sometimes the problem gives a relation between $\sin$ and $\cos$ and asks for the angle or the ratios. There are two paths: divide to get the tangent, or put everything in terms of a single ratio using the fundamental identity and solve an equation (often a quadratic).
From the value of the ratio to the angle
When you know the value of a ratio and it does not belong to a special angle, you get the angle with the inverse function on the calculator: $\sin^{-1}$ (arcsin), $\cos^{-1}$ (arccos), or $\tan^{-1}$ (arctan), with the calculator in DEG mode. If $\sin\theta = k$, the calculator returns the acute angle (or the one in quadrant I or IV); the other angle in the interval comes from the reference angle, as in the previous section.
Given the relation $5\sin \theta - 2\cos \theta = 0$, determine all possible exact values for $\sin \theta$, $\cos \theta$ and $\tan \theta$. Discuss with reasons which…
Given the relation $5\sin \theta - 2\cos \theta = 0$, determine all possible exact values for $\sin \theta$, $\cos \theta$ and $\tan \theta$. Discuss with reasons which quadrants the angle $\theta$ can lie in.
Find all angles $\alpha \in [0^\circ, 360^\circ)$ satisfying the following trigonometric equation: $2\cos^2 \alpha - 3\sin \alpha = 0$ (Hint: express the entire equation…
Find all angles $\alpha \in [0^\circ, 360^\circ)$ satisfying the following trigonometric equation:
$2\cos^2 \alpha - 3\sin \alpha = 0$
(Hint: express the entire equation in terms of a single trigonometric ratio using the fundamental identity).
A light ray passes from a liquid medium with refractive index $n_1 = 1.33$ to a crystal with index $n_2 = 1.52$. Snell's law states that…
A light ray passes from a liquid medium with refractive index $n_1 = 1.33$ to a crystal with index $n_2 = 1.52$. Snell's law states that $n_1 \sin \theta_1 = n_2 \sin \theta_2$, where $\theta_1$ and $\theta_2$ are the angles of incidence and refraction measured relative to the normal to the surface. If the ray is incident at an angle of $\theta_1 = \frac{\pi}{4}$ rad, determine the exact value of the cosine of the angle of refraction $\theta_2$.
Applications
Trigonometric ratios model any inclination: the height of a tower from its shadow, the displacement of a pendulum, the belt between two pulleys, or the area of a polygon.
A building casts a shadow of $24$ meters 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 $24$ meters 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.
A pendulum of length $80$ cm moves from its vertical equilibrium position to form an angle of $18^\circ$ with the vertical. Calculate the exact horizontal distance that…
A pendulum of length $80$ cm moves from its vertical equilibrium position to form an angle of $18^\circ$ with the vertical. Calculate the exact horizontal distance that the pendulum bob has moved and the height it has risen with respect to its lowest point. Express the results in terms of the trigonometric ratios of $18^\circ$.
Prove that the area of a regular polygon of $n$ sides inscribed in a circle of radius $R$ is given by the expression…
Prove that the area of a regular polygon of $n$ sides inscribed in a circle of radius $R$ is given by the expression $A = \frac{n \times R^2}{2} \sin\left(\frac{360^\circ}{n}\right)$. Then, apply the formula to calculate the exact area of a regular octagon inscribed in a circle of radius $10$ cm.
Two circular pulleys of radii $R = 15$ cm and $r = 5$ cm are connected by a taut belt crossed between them. The distance between the centers of the pulleys is $40$ cm.…
Two circular pulleys of radii $R = 15$ cm and $r = 5$ cm are connected by a taut belt crossed between them. The distance between the centers of the pulleys is $40$ cm. If $\alpha$ is the angle formed by the straight section of the belt with the imaginary line joining the centers, calculate the exact value of $\sin \alpha$, $\cos \alpha$ and the total length of the straight section of the belt between the two points of tangency.
Prove that if $\alpha$ and $\beta$ are two acute complementary angles (that is, $\alpha + \beta = \frac{\pi}{2}$ rad), then the following trigonometric relation always…
Prove that if $\alpha$ and $\beta$ are two acute complementary angles (that is, $\alpha + \beta = \frac{\pi}{2}$ rad), then the following trigonometric relation always holds:
$\frac{\tan \alpha + \tan \beta}{\sec \alpha \times \sec \beta} = 1$