f(x)=sinx, g(x)=cosx, and h(x)=tanx . Use the figures below to evaluate each function. g((θ)/(2)

F(x)=sinx, g(x)=cosx, and h(x)=tanx . Use the figures below...

Key Concepts

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent are fundamental in mathematics, connecting angles to ratios of sides in right triangles and coordinates on the unit circle. They have periodic behavior and are used to model waves, oscillations, and circular motion.

Function Evaluation

Function evaluation is the process of substituting a specific input into a defined function to determine its output. In the context of trigonometric functions, this involves replacing the variable (typically an angle) with a given value and computing the resulting trigonometric value.

Half-Angle Concepts

The concept of a half-angle arises when an angle is divided by two before being used in a function. Understanding half-angle relationships is important because it often leads to the use of specific half-angle identities and is crucial in simplifying trigonometric expressions and solving equations.

Unit Circle Interpretation

The unit circle is a key tool in understanding trigonometric functions, where any angle corresponds to a specific point on the circle with coordinates equal to the cosine and sine of that angle. This geometric interpretation aids in visualizing and calculating the values of trigonometric functions, including when working with fractional angles such as half-angles.

Transcript

00:01 In this problem, we're asked to evaluate what g function of theta over 2 is using this diagram.

00:09 So we know that the g function is cosine of x, so therefore i can rewrite this function as cosine of theta over 2.

00:20 So therefore, theta over 2 is a half of the angle, so therefore i'm going to use the half angle formula for cosine, which is equal to plus minus the square root of 1 plus cosine.

00:32 A theta over two.

00:34 So how do you know if you should use a negative square root instead of positive square root? well, here's my theta right here.

00:40 Okay.

00:41 So you know that your theta is going to be, you know, more than 90 degrees.

00:45 But if you take half of that, that angle is going to be somewhere in quadrant one.

00:50 So for example, you know, angles in quadrant two, you know, anything less than 180, but greater than 90.

00:57 So if you take 100 degrees, half of it, that that angle is going to be in quadrant 1.

01:03 So therefore, we want the positive square root of 1 plus cosine of theta over 2.

01:12 So to figure out what cosine of data equals to you, we're going to use the reference angle here to create a right triangle.

01:20 So that will help us to find what cosine of theta is.

01:24 So here's my right triangle.

01:25 So in this circle here, we're given the equation circle, x squared plus y equals 5, where the center is, and the radius, you have to take the square root of 5.

01:36 So i'm going to enlarge this here so you can see it.

01:40 So therefore, my hypotenuse is square to 5, and my height of the right triangle is going to be 2, because that's the y value.

01:49 But i need to find the adjacent side, which is the x -axis here.

01:54 So you can use pythaguan theorem to find that out.

01:57 So i'm going to use the pythaguan theorem 2 squared plus b squared equals to square root of 5, so that gives you 4 plus b squared equals to 5 and b squared equals to 1.

02:10 And then square root, so it's b positive 1 or negative 1...

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