Basic Trigonometric Functions

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Ratios: The Idea Behind Trig Functions

Imagine two lines that extend infinitely from one point. Let's call the the angle that these lines make \angle A.


We can draw right triangles using \angle A by creating lines that are perpendicular to one of our original two lines, as shown in Image 1. In the picture, the lines are drawn perpendicular to the side that is oriented horizontally, but they could be drawn to the other side instead without affecting our results.


Visually, we can see that  \vartriangle ABD and  \vartriangle ACE have the same three angles, and so they are similar. That is, \overline{AC} divided by \overline{AB} is the same as \overline{AE} divided by \overline{AD}:

 \frac{ \overline{AC} }{\ \overline{AB}\ }= \frac{ \overline{AE} }{\ \overline{AD}\ }


Those aren't the only corresponding pairs of sides in our diagram, though. It's also true (by the definition of similarity) that the ratio of any two sides in \vartriangle ABD is equal to the ratio of the corresponding sides in \vartriangle ACE:

 \frac{ \overline{AC} }{\ \overline{AE}\ }= \frac{ \overline{AB} }{\ \overline{AD}\ }

 \frac{ \overline{EC} }{\ \overline{AE}\ }= \frac{ \overline{DB} }{\ \overline{AD}\ }

 \frac{ \overline{EC} }{\ \overline{AC}\ }= \frac{ \overline{DB} }{\ \overline{AB}\ }

Because these ratios are the same any time you have a right triangle with a given angle, every angle can be thought of as having an associated collection of ratios. We use trigonometric functions to connect an angle with its associated ratios. Since "trigonometric" is a long word, we often shorten it to "trig".

Specific Functions

Image 2. A typical right triangle.
Image 2. A typical right triangle.
Image 3. A right triangle with base 8, height 6, hypotenuse 10, and angle 36.9°.
Image 3. A right triangle with base 8, height 6, hypotenuse 10, and angle 36.9°.

The trig functions below are defined in terms of a typical right triangle, as show in Image 2. We will then find their values for the specific triangle shown in Image 3.

Sine

Sine is a function that takes an angle and gives you the ratio of its opposite side divided by the hypotenuse:

\sin A = \frac{ \overline{CB} }{\ \overline{AC}\ }

In the triangle with angle A = 36.9°,

\sin 36.9 = \frac{6}{10} = \frac{3}{5}

Cosine

Cosine is a function that takes an angle and gives you the ratio of its adjacent side divided by the hypotenuse:

\cos A = \frac{ \overline{AB} }{\ \overline{AC}\ }

In the triangle with angle A = 36.9°,

\cos 36.9 = \frac{8}{10} = \frac{4}{5}

Tangent

Tangent is a function that takes an angle and gives you the ratio of its opposite side divided by its adjacent side:

\tan A = \frac{ \overline{CB} }{\ \overline{AB}\ }

In the triangle with angle A = 36.9°,

\tan 36.9 = \frac{6}{8} = \frac{3}{4}


Reciprocal Functions

The reciprocal trig functions are functions that have a reciprocal relationship to the "Big Three" trig functions.

Secant

Secant is a function that takes an angle and gives you the ratio of the hypotenuse divided by its adjacent side. It is the reciprocal of cosine.

\sec A = \frac{ \overline{AC} }{\ \overline{AB}\ } = \frac{1}{ \cos A}

In the triangle with angle A = 36.9°,

\sec 36.9 = \frac{10}{8} = \frac{5}{4}

Cosecant

Cosecant is a function that takes an angle and gives you the ratio of the hypotenuse divided by its opposite side. It is the reciprocal of sine.

\csc A = \frac{ \overline{AC} }{\ \overline{CB}\ } = \frac{1}{ \sin A}

In the triangle with angle A = 36.9°,

\csc 36.9 = \frac{10}{6} = \frac{5}{3}

Cotangent

Cotangent is a function that takes an angle and gives you the ratio of the its adjacent side divided by its opposite side. It is the reciprocal of tangent.

\cot A = \frac{ \overline{AB} }{\ \overline{CB}\ } = \frac{1}{ \cos A}

In the triangle with angle A = 36.9°,

\cot 36.9 = \frac{8}{6} = \frac{4}{3}


For more information on graphing these functions, see Sine Curve.

The Unit Circle


Image 4. A right triangle created within the Unit Circle.
Image 4. A right triangle created within the Unit Circle.

Up until now, we have been defining trigonometric functions in terms of right triangles. Right triangles are one of the simplest ways to begin working with trig functions, but a big problem with that definition is that we have no way to define trig functions for angles that are greater than 90 ° (or π/2 radians). If an angle is bigger than 90 °, we can't draw a right triangle with that as one of the non-right angles, and then we have no way to assign it a value for sine, cosine, or tangent.


The Unit Circle gives us a new way to define trig functions - a definition that works for all angles, even those too big to draw right triangles from.


The Unit Circle is a circle with a radius of 1 unit that is centered at the origin. In the context of the Unit Circle, angles are depicted as central angles with one side fixed on the x-axis and the other rotated counter-clockwise around the origin.


We'll start in the first quadrant, where both the x and y coordinates are positive. We can relate our right-triangle based definitions of trig functions from above to the Unit Circle by dropping a line down to the x-axis from any point on the circle. By doing so, we create a right triangle whose hypotenuse is a radius of the Unit Circle. The length of this triangle's horizontal leg is the point's x coordinate, and the length of its vertical leg is the point's y coordinate.


Before, we found an angle's sine and cosine in terms of the sides of a right triangle. Now, we will find the sides of the right triangle in terms of an angle and that angle's sine and cosine. In Image 4, x and y are the sides of the right triangle, r is the radius of the circle (in the Unit Circle, r=1), and θ is the central angle that we're looking at. Then, by our definition of sine and cosine:

\cos \theta = \frac{adjacent}{hypotenuse} = \frac{x}{r} = \frac{x}{1}


x = \cos \theta

\sin \theta = \frac{opposite}{hypotenuse} = \frac{y}{r} = \frac{y}{1}


y = \sin \theta


If the radius of our circle is not 1, the same calculations show that x=r \cos\theta and y=r \sin \theta:

\cos \theta = \frac{adjacent}{hypotenuse} = \frac{x}{r}


x =r \cos \theta

\sin \theta = \frac{opposite}{hypotenuse} = \frac{y}{r}


y = r \sin \theta


Image 5. A radius defining an obtuse angle.
Image 5. A radius defining an obtuse angle.

If x and y are related to sine and cosine, is there something in our new picture that's related to tangent? Since tangent was defined as the opposite side divided by the adjacent side,

\tan \theta= \frac{y}{x} = \frac{r \sin \theta}{r \cos \theta} = \frac{ \sin \theta}{ \cos \theta}.


We now have a new way of defining sine, cosine, and tangent. Sine and cosine are the x and y coordinates of points on the Unit Circle, and tangent is sine divided by cosine.


Now that we can define our functions in terms of a radius and an angle, we can simply orient the radius so that it makes an obtuse angle with the positive x-axis. Then the x and y coordinates of the point where the radius intersects the Unit Circle will give us the values of sine, cosine, and tangent for that angle.


In Image 5, θ is an angle greater than 90°. We are interested in finding the coordinates (x,y) of the point on the Unit Circle defined by the radius at angle θ. First, we draw a vertical line from our point to the x-axis. Next, we label the sides of our triangle. Because the point is located in the second quadrant, we know that x is negative; however since length can't be negative, the leg of our triangle along the x-axis must have length |x|.


Now our picture looks very familiar - in fact, it looks just like Image 4, but with the triangle flipped over the y-axis. This is important.


Let's label the angle inside of the triangle α. This angle is the supplementary angle of θ. Next, we draw another triangle, this one measuring α from the positive x-axis. The updated version of our picture is shown in Image 6. In this new triangle, we know how to find the coordinates of the point on the Unit Circle defined by the radius at angle α. These are the coordinates that correspond to the triangle's side-lengths, so they're just r \cos \alpha and r \sin \alpha.



Remember that we wanted to find x and y because they are r \cos \theta and r \sin \theta and we wanted to know what \cos \theta and \sin \theta are when θ is greater than 90°. Since the x and y that we're actually looking for are simply the coordinates we found for the vertex of our second triangle reflected over the y-axis, x=-r \cos \alphaand y=r \sin \alpha. This tells us that when θ is in the second quadrant,

\cos \theta = - \cos \alpha = - \cos (180 - \theta)

and

\sin \theta = \sin \alpha = \sin (180 - \theta).


Because we used the angle α to help us find out information about the angle θ, we say that α is the reference angle of θ. Whenever we work with an angle greater than 90°, we will need to find a reference angle. To find the reference angle of some obtuse angle θ, we first rotate the radius θ degrees counter-clockwise from the positive x-axis. Then, we measure the smallest angle (going either direction) between this radius and the x-axis. This angle is θ's reference angle.


When our angle is greater than 180° but less than 270°, the radius that defines it is in the third quadrant, and so both sine and cosine are negative, because both the x and y coordinates are negative. When our angle is greater than 270° but less than 360°, it's in the fourth quadrant, and only the sine (the y-coordinate) is negative. In each case, we can find cosine and sine for our angle by finding the sine and cosine of the reference angle, and then applying negative signs depending on what quadrant we're in. This process is illustrated in Image 7.


Image 7. 150°, 220°, and 330° all have a reference angle of 30°. In the first quadrant, sine and cosine are both positive. In the second quadrant, sine is positive but cosine is negative. In the third quadrant, they're both negative. In the fourth quadrant, cosine is positive but sine is negative. In each case, tangent is sine over cosine.
Image 7. 150°, 220°, and 330° all have a reference angle of 30°. In the first quadrant, sine and cosine are both positive. In the second quadrant, sine is positive but cosine is negative. In the third quadrant, they're both negative. In the fourth quadrant, cosine is positive but sine is negative. In each case, tangent is sine over cosine.


Thus, for angles greater than 90°, we have now defined the trigonometric functions in terms of a reference angle. This introduces an interesting property of trig functions: they are periodic. By starting at the positive x-axis and rotating the radius around the Unit Circle, the values of sine, cosine, and tangent cycle through a specific set of values. Once the radius reaches the positive x-axis again, these values begin to repeat.


Below is an interactive applet that demonstrates the periodic properties of the Unit Circle:


Inverse Trig Functions

Inverse trig functions can be written in two (nearly equivalent) ways - either by putting a superscripted -1 after the function name or by prefixing the function with "arc". So the inverse of \sin x is \sin^{-1} x or \arcsin x. Both are used to an equal degree. \sin^{-1} x uses function notation for the inverse of a function, however, the superscript is easily confused with an exponent, which also frequently are used with trig functions.


The concept of inverse trig functions is fairly straightforward. Just as other functions and operations have inverses (subtraction is the inverse of addition, division is the inverse of multiplication, logarithms are the inverse of exponentiation, etc.), so do trig functions. Since regular trig functions take an angle and return a ratio, their inverses take a ratio (which will be a number between -1 and 1) and return the angle to which that ratio belongs. So, \sin \left (30^\circ \right ) = \frac{1}{2} and \sin^{-1} \left ( \frac{1}{2} \right )= 30^\circ.

In this way: \sin \left (x \right ) = \frac{1}{2} \longrightarrow x=\sin^{-1} \left ( \frac{1}{2} \right )


Understanding of this section requires knowledge of radians and the unit circle.

Image 8. Note that sin-1 (the vertical graph) does not pass the vertical line test, and is, therefore, not a well defined function.  The restricted red section of sin-1 does, however, pass this test and is, therefore, a well defined function.
Image 8. Note that sin-1 (the vertical graph) does not pass the vertical line test, and is, therefore, not a well defined function. The restricted red section of sin-1 does, however, pass this test and is, therefore, a well defined function.


The input of a trig function is the output of an inverse trig function, and the output of a trig function is the input of an inverse trig function. Trig functions output ratios when given angle measure, while inverse trig functions output angle measures when given ratios. Graphically, the two relate to each other in that, the x values of the first are the y values of the second, while the y values of the first are the x values of the second. Drawing the graph of an inverse function can be accomplished simply by reflecting the graph of the original function over the line y = x, thus flipping the x and y values of the coordinates. This is illustrated in Image 8 for sine and sin-1.


As was discussed in The Unit Circle, the values of trig functions repeat every 360°. Hence, there are an infinite number of angles that will yield a specific ratio when put into a trig function. In other words, as is, the inverse trig functions do not pass the vertical line test. There is a way to remedy the problem of inverse trig functions returning an infinite number of angles for every ratio: inverse trig functions are defined with a restricted, or shortened, range from that of their corresponding trig function.

A portion that passes the vertical line test (the red portion in Image 8) is taken from each of the inverse trig functions, and is set to represent the function as a whole. Restricting the range in this way, ensures that an inverse trig function will output at most one angle for a given ratio. Note, however, that the angle that an inverse trig function outputs for a given ratio will be the reference angle for that ratio.


Inverse Trigonometric Functions
Function If Then Domain (input) of function Range (output) of function in degrees Range (output) of function in radians
sin-1 y = sin(x) x = sin-1(y) -1 ≤ x ≤ 1 -90° ≤ y ≤ 90° -π/2 ≤ y ≤ π/2
cos-1 y = cos(x) x = cos-1(y) -1 ≤ x ≤ 1 0° ≤ y ≤ 180° 0 ≤ y ≤ π
tan-1 y = tan(x) x = tan-1(y) All real numbers -90° < y < 90° -π/2 < y < π/2
csc-1 y = csc(x) x = csc-1(y) x ≤ -1 OR 1 ≤ x -90° ≤ y < 0° OR 0° < y ≤ 90° -π/2 ≤ y < 0 OR 0 < y ≤ π/2
sec-1 y = sec(x) x = sec-1(y) x ≤ -1 OR 1 ≤ x 0° ≤ y < 90° OR 90° < y ≤ 180° 0 ≤ y < π/2 OR π/2 < y ≤ π
cot-1 y = cot(x) x = cot-1(y) All real numbers 0° < y < 180° 0 < y < π


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