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Math 30-1 (grade 12) Trigonometry

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Unit 11: Trigonometry

About this unit.

Let's extend trigonometric ratios sine, cosine, and tangent into functions that are defined for all real numbers. You might be surprised at how we can use the behavior of those functions to model real-world situations involving carnival rides and planetary distances.

Unit circle introduction

• Unit circle (Opens a modal)
• The trig functions & right triangle trig ratios (Opens a modal)
• Trig unit circle review (Opens a modal)
• Unit circle Get 3 of 4 questions to level up!
• Intro to radians (Opens a modal)
• Radians & degrees (Opens a modal)
• Degrees to radians (Opens a modal)
• Radians to degrees (Opens a modal)
• Radian angles & quadrants (Opens a modal)
• Radians & degrees Get 3 of 4 questions to level up!
• Unit circle (with radians) Get 3 of 4 questions to level up!

The Pythagorean identity

• Proof of the Pythagorean trig identity (Opens a modal)
• Using the Pythagorean trig identity (Opens a modal)
• Pythagorean identity review (Opens a modal)
• Use the Pythagorean identity Get 3 of 4 questions to level up!

Trigonometric values of special angles

• Trig values of π/4 (Opens a modal)
• Trig values of special angles Get 3 of 4 questions to level up!

Graphs of sin(x), cos(x), and tan(x)

• Graph of y=sin(x) (Opens a modal)
• Intersection points of y=sin(x) and y=cos(x) (Opens a modal)
• Graph of y=tan(x) (Opens a modal)

Amplitude, midline and period

• Features of sinusoidal functions (Opens a modal)
• Midline, amplitude, and period review (Opens a modal)
• Midline of sinusoidal functions from graph Get 3 of 4 questions to level up!
• Amplitude of sinusoidal functions from graph Get 3 of 4 questions to level up!
• Period of sinusoidal functions from graph Get 3 of 4 questions to level up!

Transforming sinusoidal graphs

• Amplitude & period of sinusoidal functions from equation (Opens a modal)
• Transforming sinusoidal graphs: vertical stretch & horizontal reflection (Opens a modal)
• Transforming sinusoidal graphs: vertical & horizontal stretches (Opens a modal)
• Amplitude of sinusoidal functions from equation Get 3 of 4 questions to level up!
• Midline of sinusoidal functions from equation Get 3 of 4 questions to level up!
• Period of sinusoidal functions from equation Get 3 of 4 questions to level up!

Graphing sinusoidal functions

• Example: Graphing y=3⋅sin(½⋅x)-2 (Opens a modal)
• Example: Graphing y=-cos(π⋅x)+1.5 (Opens a modal)
• Sinusoidal function from graph (Opens a modal)
• Graph sinusoidal functions Get 3 of 4 questions to level up!
• Construct sinusoidal functions Get 3 of 4 questions to level up!
• Graph sinusoidal functions: phase shift Get 3 of 4 questions to level up!

Sinusoidal models

• Interpreting trigonometric graphs in context (Opens a modal)
• Trig word problem: modeling daily temperature (Opens a modal)
• Trig word problem: modeling annual temperature (Opens a modal)
• Trig word problem: length of day (phase shift) (Opens a modal)
• Trigonometry: FAQ (Opens a modal)
• Interpreting trigonometric graphs in context Get 3 of 4 questions to level up!
• Modeling with sinusoidal functions Get 3 of 4 questions to level up!
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Trig Calculator

The sine and cosine trigonometric functions, tan, cot, sec, and csc, calculated from trig identities., trigonometry in a right triangle, other trigonometric calculators.

Welcome to this trigonometric calculator, a trig tool created to:

• Calculate any trigonometric function by inputting the angle at which you want to evaluate it; and
• Solve for the sides or angles of right triangles by using trigonometry.

Keep reading this article to learn more about trigonometric functions and the trig identities that relate them.

Trig functions are functions that take an angle as the argument. We define these functions by using the angle of a right triangle that is inserted in a unitary circle . Then, we relate that angle to the sides of such a triangle.

As the right triangle is circumscribed in a unit circle, the length of its hypotenuse equals the circle's radius (which equals one unit).

Sine and cosine are the fundamental trigonometric functions arising from the previous diagram:

• The sine of theta ( sin θ ) is the hypotenuse's vertical projection (green line); and
• The cosine of theta ( cos θ ) is the hypotenuse's horizontal projection (blue line).

We can rotate the radial line through the four quadrants and obtain the values of the trig functions from 0 to 360 degrees , as in the diagram below:

• For example, for an angle that leads to the second quadrant (90-180°), the cosine will be negative as the horizontal projection of the hypotenuse will point to the left. That's the case of 135°, whose cosine (horizontal projection, or x coordinate) equals -√2/2.
• For the third quadrant (180-270°), the cosine and the sine of the angles lying on it will be negative.
• For the fourth quadrant angles (270-360)°, the cosines will be positive and the sines negative.

Beyond 360 degrees

The previous behavior repeats cyclically, so trigonometric functions are not limited to 360°. We can keep rotating counterclockwise, and once we reach 360 degrees, the sine and cosine functions start to repeat the same behavior. As a consequence, we can relate the functions at different angles with the following trig identities for any n integer:

• sin(θ + 2πn) = sin(θ) ;
• cos(θ + 2πn) = cos(θ); and

For example a trig function at 90° (π/2) will be mathematically the same as at 450° (5π/2), as 5π/2 = π/2 + 2π.

Negative angles

Negative angles imply the same way to calculate sine and cosine (vertical and horizontal projections, respectively), with the difference that angular rotation occurs in the clockwise direction. For example, a trigonometric function at 270° is the same as at -90°, as their radial lines are the same (you can check it with this calculator)

Once you know the value of sine and cosine, you can use the following trigonometric identities to obtain the values of the other four functions:

Tangent is the sine-to-cosine ratio

tan(α) = sin(α)/cos(α)

Cosecant is the reciprocal of the sine

csc(α) = 1/sin(α)

Secant is the reciprocal of the cosine

sec(α) = 1/cos(α)

Cotangent is the reciprocal of the tangent

cot(α) = 1/tan(α)

From the previous analysis, we can obtain some valuable formulas that relate the angle of a right triangle to its sides. Of course, this doesn't limit to unit circles, so we can use it for hypotenuses of any length.

We relate the angle of the right triangle to its sides in the following way:

sin(α) = opposite/hypotenuse

cos(α) = adjacent/hypotenuse

tan(α) = opposite/adjacent

Remember that cotangent, secant, and cosecant are the inverse of the previous functions:

csc(α) = 1/sin(α) = hypotenuse/opposite

sec(α) = 1/cos(α) = hypotenuse/adjacent

cot(α) = 1/tan(α) = adjacent/opposite

You also can apply the previous formulas for the other acute angle ( β ), but consider that the legs of the triangle will switch: the adjacent will now be the hypotenuse and vice versa.

If you like this calculator, you may find these other tools interesting:

• Trigonometry calculator ;
• Cosine triangle calculator ;
• Sine triangle calculator ;
• Trig triangle calculator ;
• Right triangle trigonometry calculator ;
• Sine cosine tangent calculator ;
• Tangent ratio calculator ; and
• Tangent angle calculator .

How do I solve a 45-90-45 degree triangle by trig formulas?

If one leg of a 45 45 90 triangle is equal to a , then:

• The second leg also equals a ;
• The hypotenuse equals a √2 (from the hypotenuse formula c = √( a ² + a ²) = a √2 );
• The area is A = a ²/2 ; and
• The perimeter equals a (2 + √2) (the sum of the two sides plus the hypotenuse).

What are the values of the 6 trig functions at 90 degrees?

The values of the 6 trig functions for 90 degrees (π/2) are the following ones:

• sin(90°) = 1;
• cos(90°) = 0;
• tan(90°)= undefined;
• cot(90°) = 0;
• sec(90°) = undefined; and
• csc(90°) = 1.

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Introduction to Trigonometry

Trigonometry (from Greek trigonon "triangle" + metron "measure")

Want to learn Trigonometry? Here is a quick summary. Follow the links for more, or go to Trigonometry Index

Trigonometry helps us find angles and distances, and is used a lot in science, engineering, video games, and more!

Right-Angled Triangle

The triangle of most interest is the right-angled triangle . The right angle is shown by the little box in the corner:

Another angle is often labeled θ , and the three sides are then called:

• Adjacent : adjacent (next to) the angle θ
• Opposite : opposite the angle θ
• and the longest side is the Hypotenuse

Why a Right-Angled Triangle?

Why is this triangle so important?

Imagine we can measure along and up but want to know the direct distance and angle:

Trigonometry can find that missing angle and distance.

Or maybe we have a distance and angle and need to "plot the dot" along and up:

Questions like these are common in engineering, computer animation and more.

And trigonometry gives the answers!

Sine, Cosine and Tangent

The main functions in trigonometry are Sine, Cosine and Tangent

They are simply one side of a right-angled triangle divided by another.

For any angle " θ ":

(Sine, Cosine and Tangent are often abbreviated to sin, cos and tan .)

Example: What is the sine of 35°?

Using this triangle (lengths are only to one decimal place):

sin(35°) = Opposite Hypotenuse = 2.8 4.9 = 0.57...

The triangle could be larger, smaller or turned around, but that angle will always have that ratio .

Calculators have sin, cos and tan to help us, so let's see how to use them:

Example: How Tall is The Tree?

We can't reach the top of the tree, so we walk away and measure an angle (using a protractor) and distance (using a laser):

• We know the Hypotenuse
• And we want to know the Opposite

Sine is the ratio of Opposite / Hypotenuse :

sin(45°) = Opposite Hypotenuse

Get a calculator, type in "45", then the "sin" key:

sin(45°) = 0.7071...

What does the 0.7071... mean? It is the ratio of the side lengths, so the Opposite is about 0.7071 times as long as the Hypotenuse.

We can now put 0.7071... in place of sin(45°):

0.7071... = Opposite Hypotenuse

And we also know the hypotenuse is 20 :

0.7071... = Opposite 20

To solve, first multiply both sides by 20:

20 × 0.7071... = Opposite

Opposite = 14.14m (to 2 decimals)

The tree is 14.14m tall

Try Sin Cos and Tan

Play with this for a while (move the mouse around) and get familiar with values of sine, cosine and tangent for different angles, such as 0°, 30°, 45°, 60° and 90°.

Also try 120°, 135°, 180°, 240°, 270° etc, and notice that positions can be positive or negative by the rules of Cartesian coordinates , so the sine, cosine and tangent change between positive and negative also.

So trigonometry is also about circles !

Unit Circle

What you just played with is the Unit Circle .

It is a circle with a radius of 1 with its center at 0.

Because the radius is 1, we can directly measure sine, cosine and tangent.

Here we see the sine function being made by the unit circle:

Note: you can see the nice graphs made by sine, cosine and tangent .

Degrees and Radians

Angles can be in Degrees or Radians . Here are some examples:

Repeating Pattern

Because the angle is rotating around and around the circle the Sine, Cosine and Tangent functions repeat once every full rotation (see Amplitude, Period, Phase Shift and Frequency ).

When we want to calculate the function for an angle larger than a full rotation of 360° (2 π radians) we subtract as many full rotations as needed to bring it back below 360° (2 π radians):

Example: what is the cosine of 370°?

370° is greater than 360° so let us subtract 360°

370° − 360° = 10°

cos(370°) = cos(10°) = 0.985 (to 3 decimal places)

And when the angle is less than zero, just add full rotations.

Example: what is the sine of −3 radians?

−3 is less than 0 so let us add 2 π radians

−3 + 2 π = −3 + 6.283... = 3.283... rad ians

sin(−3) = sin(3.283...) = −0.141 (to 3 decimal places)

Solving Triangles

Trigonometry is also useful for general triangles, not just right-angled ones .

It helps us in Solving Triangles . "Solving" means finding missing sides and angles.

Example: Find the Missing Angle "C"

Angle C can be found using angles of a triangle add to 180° :

So C = 180° − 76° − 34° = 70°

We can also find missing side lengths. The general rule is:

When we know any 3 of the sides or angles we can find the other 3 (except for the three angles case)

See Solving Triangles for more details.

Other Functions (Cotangent, Secant, Cosecant)

Similar to Sine, Cosine and Tangent, there are three other trigonometric functions which are made by dividing one side by another:

Trigonometric and Triangle Identities

And as you get better at Trigonometry you can learn these:

Enjoy becoming a triangle (and circle) expert!

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Reference Angles & Trig Values

There are a few (a very few) angles that spit out relatively "neat" trigonometric values, involving, at worst, one square root. Because of their relatively simple values, these are the angles which will typically be used in math problems (in calculus, especially), and you will be expected to know these value by heart.

Usually, textbooks present this short list of values in a table that you are expected to memorize. But pictures are generally easier to recall (on tests, etc) than tables, so this lesson will show the way in which many people really keep track of these values.

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45-45-90 and 30-60-90 Triangles

How do I find the trig values for 45° angles?

To find (or recall) the trig values for 45° angles:

• Draw a 45-45-90 triangle with short sides having lengths of either (a) sqrt(2) or (b) 1 .
• Label the hypotenuse as having a corresponding length of either (a) 2 or (b) sqrt(2) .
• From whichever triangle you have drawn (either Triangle (a) or Triangle (b)), read off the trig ratio you need.

← swipe , if needed, to view full table →

Why are there two different ways of setting up this triangle? Because some instructors don't want any square roots in the denominators for 45° -angle trig values; for that instructor, you'd use Triangle (a). But other instructors, and certainly those in later courses (like calculus) won't care about radicals in the denominators; they'd prefer you use the "simplified" ratios generated by Triangle (b). (I use Triangle (b) in my own computations.) So let's do the two cases. (In each case, the angle symbol used, which looks in the pictures above kind of like a zero with a line across it, is the Greek letter theta, pronounced "THAY-tuh".)

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Triangle (a) is one whose values you'll just have to memorize, if it's the one you're supposed to be using (that is, if you're not allowed to have radicals in the denominator). Label the two matching legs as being of length sqrt(2) . Then the Pythagorean Theorem says that the hypotenuse must have a length of 2 . Then you can read off the trigonometric values you need.

For instance, if you needed to find the tangent (which is "opposite over adjacent"), you would form the ratio sqrt(2)/sqrt(2) = 1 , which is the correct value for the tangent. The values of the sine and cosine for a 45° angle are the same; sin (θ) = sqrt(2)/2 and cos (θ) = sqrt(2)/2 . If you need to find one of the co-functions (such as cosecant), you may be required to rationalize some denominators .

Triangle (b) is, in my experience, the easier one to work with. For a start, the equal-length legs have a length of one unit — this is a simpler value — and then we get a length of sqrt(2) for the hypotenuse by using the Pythagorean Theorem .

If, for a given exercise, they only want the trigonometric value of the angle, then just read it off the triangle: the sine of θ is (opposite) over (hypoteneuse), or 1/sqrt(2) ; the cosine of theta is (adjacent) over (hypoteneuse), or 1/sqrt(2) ; and the tangent of theta is (opposite) over (adjacent), or 1/1 = 1 .

Note: It is irrelevant what are the lengths of the actual triangle you are dealing with when you're needing to find trig values; the reference triangle you use will give you the necessary and correct ratios, and thus the correct trigonometric values. I guess the point here is that, whatever is the fraction or ratio that you get from a particular 45° triangle, it will always simplify down to what you get from the reference Triangles above. You can find anything you need from the reference Triangle of your choice. If trying to memorize an entire table isn't working for you, instead simply memorize the triangle of your choice.

And don't worry if your triangle (as printed in the textbook, say, when you're doing your homework) isn't oriented in the same way. The trigonometric values for the 45° angle will always be the same. (If you've gotten as far as working with angles in other quadrants, then use the reference triangle to find the necessary value, and then do a sign change, if necessary.)

How do I find the trig values for 30° - and 60° -angles?

To find (or recall) the trig values for 30° and 60° angles:

• Draw an equilateral triangle having sides of length 2 .
• Drop the altitude line from the peak to the midpoint of the base, forming two smaller triangles.
• Relabel the base as now have two lengths of 1 .
• Label the various angles and the altitude with their values.
• From the appropriate triangle, read off the trig ratio you need.

Now let's flesh out those steps a bit, shall we? For either of the 30° - and 60° -angles, the triangle below is what you start with:

This is a 60-60-60 triangle (that is, an equilateral triangle), with all sides having a length of two units.

Drop the vertical bisector from the top angle down to the bottom side. Note that this bisector is also the altitude (that is, the height) of the triangle.

The Pythagorean Theorem tells us that the length of the bisector is sqrt(3) . Because the bisector, by definition, cut the angle at the peak in half, it has thus formed two 30-60-90 triangles.

If you are working with a 60° triangle, use the angle labelled above using the Greek letter "alpha" (the funny-looking " a " in the lower corner); if you are doing a 30 -degree triangle, use the angle labelled with the Greek letter "beta" (the funny-looking " b " in the upper corner).

In other words, for working with 60° angles, your picture is this half of the triangle:

...and for 30 -degree angles, your picture is rotated as shown below:

You find trigonometric values and ratios with the 30° and 60° triangles in the exact same manner as with the 45° triangle (ya know: SOH CAH TOA ).

What if I'm not allowed to draw pictures to find the angle values?

If your instructor doesn't want you drawing pictures to help you remember the trigonometric values, then... don't let your instructor know that you're drawing the pictures. Use scratch paper that you don't hand in, or draw on the test with something that is cleanly erasable; plus, if you can, write over that area after erasure with something else, to further obfuscate.

( Obfuscation is actually a "real world" skill, in various incarnations.)

It may be that your instructor feels that you're supposed to have everything memorized by now, so pictures are banned. Well, this is why your pencil has an eraser. My Calculus II instructor (an otherwise righteous dude) said that if we drew one of those pictures when working a question on our tests, the entire question would be counted wrong; draw a picture, get a zero. I drew the pictures anyway, but very lightly, in pencil, and erased them all before I handed the tests in. I might do the picture-drawing scratch-work for Question 12 in the space for Question 13, which I'd later overwrite (after erasure). He never knew the difference, and I passed the course. You do what you gotta do.

A helpful (shorthand) table of values

While many people like pictures, some people do prefer tables and charts. If tables work better for you, then this table comes highly recommended, having been "field-tested" by a working instructor:

← swipe , as needed, to view full table →

Finding angle values on your hand

If you don't mind your classmates watching you count on your fingers (and why should you?), then there is a way to find the values of 0°, 30°, 45°, 60° and 90° angles on your hand. I won't reinvent the wheel so, if you'd like to use that method, then a guy going by the handle "StephenwithaPhD" has an article here . (If you ever take a class where you have to multiply things called "vectors", the whole class will be waving their hands in the air for the Right-Hand Rule , so don't be shy if you want to use your fingers now!)

You can use the Mathway widget below to check your work when finding exact values for trigonometric ratios. Try the entered exercise, or type in your own exercise. Then click the button to compare your answer to Mathway's.

Please accept "preferences" cookies in order to enable this widget.

(Click here to be taken directly to the Mathway site, if you'd like to check out their software or get further info.)

URL: https://www.purplemath.com/modules/trig.htm

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• prove\:\tan^2(x)-\sin^2(x)=\tan^2(x)\sin^2(x)
• prove\:\cot(2x)=\frac{1-\tan^2(x)}{2\tan(x)}
• prove\:\csc(2x)=\frac{\sec(x)}{2\sin(x)}
• prove\:\frac{\sin(3x)+\sin(7x)}{\cos(3x)-\cos(7x)}=\cot(2x)
• prove\:\frac{\csc(\theta)+\cot(\theta)}{\tan(\theta)+\sin(\theta)}=\cot(\theta)\csc(\theta)
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• I know what you did last summer…Trigonometric Proofs To prove a trigonometric identity you have to show that one side of the equation can be transformed into the other...

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