WEBVTT mathematics/trigonometry/murray
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Hi welcome back to the trigonometry lectures on educator.com
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Today, we're going to learn about the last trigonometric functions.
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We've learned about the sine and cosine, and tangent and cotangent.
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Today we're going to learn the secant and cosecant function.
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Let's start with their definitions.
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The secant function is just defined by the sec(θ)=1/cos(θ).
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That only works when the cos(θ) is non-zero.
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If the (cosθ)=0, we just say the secant is undefined.
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The cosecant, which people shorten to csc, is just 1/sin(θ).
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If the sin(θ)=0, we just say that the cosecant is undefined.
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Let's start with the first example right away.
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We have to draw a graph of the secant function.
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In particular, we have to label all zeros, max's, mins, and asymptotes and figure what the period of the secant function is.
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Remember now that the secant function, sec(θ), by definition is 1/cos(θ).
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A really good place to start here when you're trying to understand sec(θ) is with the graph of cos(θ).
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Let me start with the graph of cos(θ).
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There's π, there's 2π, and I'm going to extend this out a bit, 3π, and -π.
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Now, remember that the cosine function starts at 1, goes down to 0 at π/2, so there's π/2, goes down to -1 at π, comes back to 0 at 3π/2, back up to 1 at 2π.
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The period of cosine is 2π, so it's repeating itself after 2π.
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I'm not drawing secant yet, I'm drawing cos(x), y=cos(x).
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In black here, I've got, we'll call it cos(θ).
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Now, I'm going to draw the secant function in red.
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That means we're doing 1/cos(x), or 1/cos(θ).
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In particular, 1/1 is 1.
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Then as the cosine goes down to 0, secant is 1/cos, so it goes up to infinity there, and does the same thing on the other side, so secant looks like that.
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When the cosine is negative, when the cosine is -1, secant is -1.
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When the cosine goes to 0, secant blows up but since cosine is negative here, secant goes to negative infinity when the cosine is negative.
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Then when cosine is positive again, secant is positive again, going up to positive infinity on both ends there.
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That's what the secant function looks like.
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While the cosine goes between -1 and 1, secant is just the reciprocal of that, so it goes 1 up to negative infinity, and from -1 down to negative infinity.
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Now, I've got the secant graph in red.
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Let me label all the things that we've been asked to label.
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First of all, zeros, while the sec(θ) has no zeros because it never crosses the x-axis, so there are no zeros to label.
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Max's, all local max's of the secant function, well here's one down here at (π,-1) is a local max.
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Minimum at (0,1) is a local min, (2π,1) is a local min, and so on, as a local min and a local max, every π units.
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We've got the max's and the mins.
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The asymptotes are the places where the secant blows up to infinity or drops down to negative infinity and so that's an asymptote at -π/2, and at π/2.
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We have an asymptote at π/2, and again at 3π/2, and again at 5π/2.
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Basically, every π units, we have an asymptote where the secant function blows up to infinity or drops down to negative infinity.
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Finally, what is the period of secant function.
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The secant function is really dependent on the cosine function, and the cosine function repeats itself once every 2π.
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The period of cosine is 2π.
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The period of secant is also 2π.
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You can see that from the graph, it starts repeating itself after a multiple of 2π.
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That's what the graph of the secant function looks like.
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It kind of has this hues and this upside down hues really dependent on the cosine function because the secant is just 1/cos(θ).
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For the second example, we have to figure out some common values of secant and cosecant at the angles in the first quadrant, 0, π/6, π/4, π/3, and π/2.
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Now, these angles are probably so common that you really should have memorized the sine and cosine.
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I'm going to start by writing down the sine and cosine of these values.
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I'm going to write them down both in degrees and radians because it's very important to be able to identify these common values either way.
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I'll write down degrees, radians, I'll write down the cosine and the sine, and then the secant and cosecant of each one.
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I'll make a nice chart here.
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The values given were 0, π/6, π/4, π/3, and π/2, in terms of degrees, that's 0, 30, 45, 60, and 90.
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Now, you really should have probably memorized the cosine and sine of these already.
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You probably shouldn't even have to check the unit circle.
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But if you need to, go ahead and draw yourself a unit circle.
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Then draw out those triangles in the first quadrant, and you'll be able to figure out the cosine and sine very quickly as long as you remember the values of the 30-60-90 triangles and the 45-45-90 triangles.
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In particular, the cosine and sine of 0, are 1 and 0.
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For 30-degree angle, cosine is root 3 over 2, sine is 1/2.
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For 45-degree angle, they're both root 2 over 2.
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For 60-degree angle, they're just the opposite of what they were for 30, 1/2 and root 3 over 2.
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For 90, they're just the opposite of what they were for 0, 0 and 1.
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Now, the secant is just the reciprocal of cosine, it's just 1/cos.
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I'll just take 1/1, 2 divided by root 3, if you rationalize that, you get 2 root 3 over 3, 2 divided by root 2, is just root 2, the reciprocal of 1/2 is 2, and the reciprocal of 0 is undefined.
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Those are the secants of those common values.
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The cosecant is 1/sin, while 1/0 is undefined, 1/(1/2) is 2.
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The reciprocal of root 2 over 2 is 2 divided by root 2, which again is root 2.
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Then 2 divided by root 3 rationalizes into 2 root 3 over 3, and then 1/1 is just 1.
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I've been really trying to drill you on memorizing the values of sine and cosine.
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I don't think it's really worth memorizing the values of secant and cosecant, they don't come up as often as sine and cosine.
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The key thing to remember is that sec(θ) is just 1/cos(θ), and csc(θ) is just 1/sin(θ).
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As long as you really have memorized the values of sine and cosine, you can always work out the values of secant and cosecant.
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I don't think you need to memorize these values of secant and cosecant.
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It helps if you practice them but it's not really worth memorizing them as long as you know your sine and cosine really well.
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You can always work out secant and cosecant.
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Last thing this example asked is, which other quadrants the secant and cosecant are positive?
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Let's go back and remember our little mnemonic here, All Students Take Calculus.
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That's in quadrant 1, quadrant 2, quadrant 3, and quadrant 4.
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That tells us which of the common functions are positive in which quadrant.
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In the first quadrant, they're all positive, in the second quadrant, only sine, in the third quadrant, only tangent, and in the fourth quadrant, only cosine.
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Let's figure out what that means for secant and cosecant in each case.
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On the first quadrant, they're both positive, both sine and cosine are positive, so secant and cosecant are both positive.
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In the second quadrant, sine is positive, which means that secant is positive, but cosine is negative so secant is negative.
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In the third quadrant, tangent is the only thing that's positive, sine and cosine are both negative, so secant and cosecant are both negative.
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Finally, in the fourth quadrant, cosine is positive so secant is positive, sine is negative so cosecant is negative.
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Now, we have a little chart that tells us which quadrant secant and cosecant are positive and negative in.
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Again, I don't think you really need to memorize this as long as you remember very well where sine and cosine are positive, you can always work out where secant and cosecant are positive and negative.
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For our next example, we're asked to find whether the secant and cosecant functions are odd, even or neither.
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Let's remember what the definition of odd and even are.
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Odd is where f(-x)=-f(x), and that also, by looking at the graph, you can identify odd functions, they have rotational symmetry around the origin.
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Even functions, f(-x)=f(x), and they have mirror symmetry across the y-axis.
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Now, let's look at sec(x), sec(x), actually we have to look at sec(-x) to check whether it's odd or even.
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So, sec(-x), secant remember is 1/cos, 1/cos(-x), cosine is an even function, so this is just 1/cos(x), which is sec(x) again, sec(x) is even.
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Csc(x), well csc(-x), cosecant is 1/sin, so that's 1/sin(-x), but sine's an odd function, so this is 1/-sin(x), which is -csc(x), so cosecant is odd.
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That was just a matter of checking the definitions of odd and even, plugging -x into secant, cosecant, and seeing what we came up with.
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We can also figure it out from the graphs if we remember what those look like.
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Secant, remember ...
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Let me draw a cosine.
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Secant is 1/cos, so that was the one that look like this, that's sec(x) in red there.
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If you look, it has mirror symmetry across the y-axis, which checks that sec(x) is really an even function.
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Mirror symmetry across the y-axis.
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Csc(x), let me draw a quick graph of csc(x).
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Remember, that's based on the graph of sin(x), so start by drawing a graph of sin(x).
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Now, we'll fill in a graph of csc(x), it has asymptotes wherever sine is 0, so that's now our graph of csc(x).
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Clearly, that does not have mirror symmetry across the y-axis, but it does have rotational symmetry around the origin.
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If you spun that around 180 degrees, it would look the same, so it does have rotational symmetry.
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That confirms that cosecant is an odd function.
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We'll try some more examples later.
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You should try working them out yourself, then we'll work them out together.