Let tan θ=(5)/(12) with θin QI and find the following. sec2...

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that relate the trigonometric functions to one another. They provide the tools needed to express one function in terms of another, such as using the identity 1 + tan²? = sec²? or the Pythagorean identity to relate sine and cosine, which is crucial for solving problems involving different trigonometric functions.

Double-Angle Formulas

Double-angle formulas express trigonometric functions of a double angle in terms of the functions of the original angle. For cosine, the formulas such as cos 2? = cos²? - sin²? or cos 2? = 1 - 2 sin²? allow one to calculate the cosine of twice an angle when the sine or cosine of the original angle is known. These formulas are fundamental when dealing with expressions involving functions of multiple angles.

Reciprocal Relationships

Reciprocal relationships in trigonometry, such as sec ? being the reciprocal of cos ? (sec ? = 1/cos ?), are important for converting one trigonometric function into another. This concept is especially useful when a problem requires finding one function based on the value of its reciprocal, as is the case when calculating sec 2? from a known cosine of 2?.

Transcript

00:01 So we're given the tangent of theta is equal to 512s.

00:05 And we know that theta is an angle in quadrant 1, and we want to find what the secant of 2 theta is.

00:15 So we know we're going to have to end up using a double angle identity.

00:19 And if we think the tangent function, we know the tangent function is the ratio.

00:26 It's basically the slope.

00:27 So we would know that this would have a quote, slope, if we draw that ray in the first quadrant, that this is angled theta, that the ratio of y over x is ending up equaling 5 over 12.

00:42 Now, we can't go through and say, hey, the sine of the tangent of theta is equal to sign of the angle over cosine of the angle.

00:53 And a lot of my students will say, oh, then the sign is 5 and the cosine is 12.

00:57 Well, that can't be because a sign can never be bigger than one and the cosine can never be bigger than one.

01:04 So this is not the sign of the angle and this is not the cosine of the angle.

01:08 So what we're gonna need to do, and there are infinitely many pairs of fractions that we could have like five over 10 and then have five, 12 over, well, couldn't be 10, but we'd have infinitely many fractions that if we reduce it down, we get five over 12.

01:26 So we need to use an identity.

01:28 And i know that the tangent squared of an angle, and i'm going to write it this way, plus one is equal to the secant squared of the angle.

01:38 And i'll write it again that way.

01:40 Now, this is not equivalent to that.

01:43 That would make the problem a lot easier if we just had to find the secant.

01:47 But we know that the tangent of the angle is 512, so we square it.

01:51 We get 25 over 144, plus 1, which will write as 144 over 144, is equal to this secant of the angle.

02:02 Theta squared.

02:04 So we know when we add these two together we get 169 over 144 and that's equal to the secant squared of the angle and we know that the square root of that would equal the secant and we know that all of our trig functions are positive in this first quadrant so we don't have to put plus or minus.

02:24 So we know the secant of the angle is equal to 13 over 12 which means the cosine of the angle is equal to 12 over 13.

02:36 And so we can go through and work with the double angle theorem for cosine to help us find this secant of 2 theta.

02:46 So here's one thing we needed.

02:49 We needed to know this or this...

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