Trigonometric identities Find the exact value of tan(3 π/ 8)...

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental equations that relate the trigonometric functions to one another. They are vital tools in simplifying expressions, proving mathematical statements, and aiding in the computation of exact trigonometric values.

Tangent Function

The tangent function is defined as the ratio of the sine and cosine functions. It plays a key role in trigonometry, particularly in relating angles to slopes and solving problems that involve angle measures, periodicity, and symmetry.

Angle Sum and Difference Identities

Angle sum and difference identities provide formulas to express trigonometric functions of the sum or difference of two angles in terms of the functions of the individual angles. These identities are essential for breaking down more complex angle measures into simpler, known values.

Half-Angle Formulas

Half-angle formulas are a subset of trigonometric identities used to determine the trigonometric functions of half of a given angle. They are particularly useful when the angle of interest does not directly correspond to a standard angle with a known value.

Exact Value Computation

Exact value computation involves using trigonometric identities and known values of trigonometric functions to determine precise, symbolic expressions for trigonometric functions at specific angles rather than relying on approximations.

Transcript

00:01 So let's see the concept of half angles, right? you know, when you have something like tangent of 3 pi over 8, 3 pi over 8, 3 pi over 8 can be written as 1 half, 3 pi, let me just make it times 3 pi over 4, right? so this is 1 half of this one, right? so you can use the concept of half angles right okay so if you're using a concept of half angles right then you know that the angle identities for half angle identities for cosine cosine square theta is actually one plus cosine two theta over two right over two so what is it going to be it's going to be one plus you know cosine three over four right because you're not using this one as your theta right this is the half that guarantee that we use the half angles definition so our theta is going to be this one right so it's going to be cosine three pi over four and then over two right you have this one so what is that going to be you well if we evaluate three pilot of a four from the unit circle and then you add it is going to be two minus square root of two over four just evaluate this one on the unit circle and then you're going to have you're going to have it right so that is it and then sign square theta is also going to be one minus cosine two theta over two right so what is that one going to be be that one is also going to be one minus cosine two three pi over four and then over two right and when you evaluate this one in a unit circle and then make it one minus that over two what do you get you get the same thing it's just that this one is going to change the sign is going to change right so this one is going to have two plus square root of two over four see so once you have that you have sign you have cosine right all you got to do is find tangent right so what is tangent tangent of any angle is actually sign of that angle over cosine of that angle right remember so we need sign of three power over eight and sign of three power over eight is actually this one, right? but that is sign squared.

03:54 Do not forget, it is sign squared...

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