You are riding a Ferris wheel. How high is your seat after the wheel has turned $210^\circ$ from the 3 o'clock position? Once you pass the top, there is no right triangle to draw. In this section you will extend sine, cosine, and tangent to any angle, learn the exact values worth memorizing, and use them to simplify, prove, and solve.
Degrees and radians
You already measure angles in degrees. Math beyond Geometry (and every calculus class) prefers radians, because they come straight from the circle: one radian is the angle whose arc on a circle has the same length as the radius.
A full turn is $360^\circ$, and its arc is the whole circumference $2\pi r$, so a full turn is $2\pi$ radians. That gives the one fact you need:
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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 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 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 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 by a particle on its outer edge in $0.15$ seconds.
In a right triangle, the ratios of the sides depend only on the acute angle. Name the sides from the angle's point of view: opposite (across from it), adjacent (touching it, not the hypotenuse), and the hypotenuse (across from the right angle, always the longest side).
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Calculate the exact value of the sine, cosine and tangent of the smallest angle of a right triangle knowing that its legs measure $5$ inches and $12$ inches.
Calculate the exact value of the sine, cosine and tangent of the smallest angle of a right triangle knowing that its legs measure $5$ inches and $12$ inches.
To optimize solar energy capture in winter, a rectangular panel $2$ feet long is raised by resting one of its shorter sides on the flat ground, so that the opposite side…
To optimize solar energy capture in winter, a rectangular panel $2$ feet long is raised by resting one of its shorter sides on the flat ground, so that the opposite side is exactly $1.2$ feet high. Determine the exact value of the cosine and tangent of the angle of inclination of the panel.
A triangle only gives acute angles. To handle $150^\circ$, $210^\circ$, or $-45^\circ$, place the angle in standard position: vertex at the origin, initial side along the positive $x$-axis, and the terminal side rotated counterclockwise for a positive angle (clockwise for a negative one).
Put the point $P$ where the terminal side meets the unit circle (radius $1$). The right triangle drawn from $P$ down to the $x$-axis has hypotenuse $1$, so its legs are exactly the coordinates of $P$.
Signs by quadrant
Sine is the $y$-coordinate, cosine is the $x$-coordinate, and tangent is $y/x$. So the signs depend only on the quadrant. The reciprocal ratios share the sign of their partner ($\csc$ with $\sin$, $\sec$ with $\cos$, $\cot$ with $\tan$).
Angles on the axes
On the axes one coordinate is $0$ and the other is $\pm 1$. You can read these values straight off the unit circle.
$\theta$
$0^\circ$
$90^\circ$
$180^\circ$
$270^\circ$
$360^\circ$
$\sin\theta$
$0$
$1$
$0$
$-1$
$0$
$\cos\theta$
$1$
$0$
$-1$
$0$
$1$
$\tan\theta$
$0$
undefined
$0$
undefined
$0$
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Determine in which quadrant the angle $\beta$ is located if it is simultaneously true that $\sin \beta < 0$ and $\tan \beta > 0$.
Determine in which quadrant the angle $\beta$ is located if it is simultaneously true that $\sin \beta < 0$ and $\tan \beta > 0$.
Special angles: $30^\circ$, $45^\circ$, and $60^\circ$
Three angles have exact ratios involving square roots. They come from two triangles you know from Geometry: the $45^\circ$-$45^\circ$-$90^\circ$ triangle (legs $1$, hypotenuse $\sqrt2$) and the $30^\circ$-$60^\circ$-$90^\circ$ triangle (half of an equilateral triangle with side $2$: sides $1$, $\sqrt3$, $2$).
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Evaluate the following expression by substituting the trigonometric ratios with their exact values, without using a calculator:…
Evaluate the following expression by substituting the trigonometric ratios with their exact values, without using a calculator:
Angles that end on the same terminal side are coterminal: they differ by whole turns ($360^\circ$ or $2\pi$). So they have identical ratios. To find the quadrant of a big or negative angle, subtract (or add) full turns until it lands in $[0^\circ, 360^\circ)$, that is, $[0, 2\pi)$.
The reference angle $\theta'$ is the acute angle between the terminal side and the $x$-axis. Every angle has the same ratios as its reference angle, except possibly for the sign.
Angles written as $\pi \pm x$ or $\frac{\pi}{2} \pm x$
Expressions like $\sin(\pi - x)$ appear when you simplify. Treat $x$ as a small acute angle, find the quadrant of the whole angle to get the sign, and find its reference angle to get the function. When the reference angle is $\frac{\pi}{2} - x$ (angles near the vertical axis), sine and cosine swap.
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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 its…
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 its cotangent.
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 this quadrant:
The point $P(\cos\theta, \sin\theta)$ is always $1$ unit from the origin. The distance formula (the Pythagorean Theorem) gives the most used identity in trigonometry.
When you solve for a ratio you get a $\pm$ square root. The identity does not choose the sign, the quadrant does.
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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 the 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 the fundamental trigonometric identities.
Knowing that $\tan \gamma = -\frac{12}{5}$ and that $\gamma$ belongs to the second quadrant, calculate the exact value of $\sin \gamma$ and $\sec \gamma$.
Knowing that $\tan \gamma = -\frac{12}{5}$ and that $\gamma$ belongs to the second quadrant, calculate the exact value of $\sin \gamma$ and $\sec \gamma$.
An identity is true for every angle where both sides are defined. To simplify an expression or prove an identity, rewrite it using the basic relationships until it becomes something simple (or until both sides match).
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Simplify the following trigonometric expression as much as possible using the fundamental identities, expressing the result in terms of a single trigonometric ratio:…
Simplify the following trigonometric expression as much as possible using the fundamental identities, expressing the result in terms of a single trigonometric ratio:
Prove the following trigonometric identity, indicating the algebraic steps and the fundamental relationships used. Assume that $\alpha$ is an angle for which all ratios…
Prove the following trigonometric identity, indicating the algebraic steps and the fundamental relationships used. Assume that $\alpha$ is an angle for which all ratios are defined:
Prove that if $\alpha$ and $\beta$ are two complementary acute angles (i.e., $\alpha + \beta = \frac{\pi}{2}$ rad), then the following trigonometric relationship is…
Prove that if $\alpha$ and $\beta$ are two complementary acute angles (i.e., $\alpha + \beta = \frac{\pi}{2}$ rad), then the following trigonometric relationship is always verified:
The same tools solve equations. If an equation mixes $\sin$ and $\cos$, either divide to get a tangent or use $\sin^2 + \cos^2 = 1$ to leave a single ratio. Then solve for that ratio and use reference angles to find every angle in the requested interval.
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Given the relationship $5\sin \theta - 2\cos \theta = 0$, determine all possible exact values for $\sin \theta$, $\cos \theta$ and $\tan \theta$. Discuss with reasoning…
Given the relationship $5\sin \theta - 2\cos \theta = 0$, determine all possible exact values for $\sin \theta$, $\cos \theta$ and $\tan \theta$. Discuss with reasoning in which quadrants the angle $\theta$ can be found.
Find all angles $\alpha \in [0^\circ, 360^\circ)$ that satisfy the following trigonometric equation: $2\cos^2 \alpha - 3\sin \alpha = 0$ (Hint: express the entire…
Find all angles $\alpha \in [0^\circ, 360^\circ)$ that satisfy 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 ray of light 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 ray of light 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 with respect 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 $\theta_2$.
An angle of elevation is measured up from the horizontal. Draw the right triangle, label the sides relative to the angle, pick the ratio that links what you know with what you want, and write the equation before using the calculator.
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A building casts a shadow of $24$ feet when the angle of elevation of the sun above the horizon is $53^\circ$. Write the trigonometric expression to calculate the height…
A building casts a shadow of $24$ feet 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 nonlinearity of the tangent ratio.
A pendulum of length $80$ inches is displaced from its vertical equilibrium position until it forms an angle of $18^\circ$ with the vertical. Calculate the exact…
A pendulum of length $80$ inches is displaced from its vertical equilibrium position until it forms an angle of $18^\circ$ with the vertical. Calculate the exact horizontal distance the mass of the pendulum has been displaced and the height it has risen with respect to its lowest point. Express the results in terms of the trigonometric ratios of $18^\circ$.
Triangles hide inside circles. Cutting a regular polygon into triangles from the center, or drawing a radius to a tangent point, gives right triangles and isosceles triangles where sine and cosine do the work.
Belts around pulleys
Draw the radius to each tangent point. A radius is perpendicular to the tangent, so both radii are perpendicular to the belt. Slide the segment from the small center parallel to the belt until it meets the big radius. That cuts off a piece of the big radius of length $r$ (next to the belt) and leaves $R - r$ from the big center. This forms a right triangle with hypotenuse $d$ (the distance between centers).
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Show that the area of a regular polygon of $n$ sides inscribed in a circle of radius $R$ is given by the expression…
Show that the area of a regular polygon of $n$ sides inscribed in a circle of radius $R$ is given by the expression $A = \frac{n \cdot R^2}{2} \sin\left(\frac{360^\circ}{n}\right)$. Then, apply the formula to calculate the exact area of a regular octagon inscribed in a circle of radius $10$ inches.
Two circular pulleys of radii $R = 15$ inches and $r = 5$ inches are connected by a tight belt that crosses between them. The distance between the centers of the pulleys…
Two circular pulleys of radii $R = 15$ inches and $r = 5$ inches are connected by a tight belt that crosses between them. The distance between the centers of the pulleys is $40$ inches. 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.
Back to the Ferris wheel. Suppose the wheel has radius $50$ ft and its bottom is $10$ ft above the ground, so the center is $60$ ft up. Your seat starts at the 3 o'clock position.
Summary
Let's practice! 8 medium exercises in this section