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

Name: Problem 32, Chapter 8: Algebra and Trigonometry
Uploaded: 2021-07-24T19:52:52-08:00
Duration: 2 min 44 s
Channel: Ziya Ogron
Description: f(x)=sinx, g(x)=cosx, and h(x)=tanx . Use the figures below to evaluate each function. g(2 θ)

step 1 For this question, we're asked to evaluate what G of 2 Theta is. So we know what G function equa

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

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Key Concepts

Cosine Function

The cosine function is one of the primary trigonometric functions and is typically used to represent the ratio of the adjacent side to the hypotenuse in a right triangle. Its properties such as periodicity, symmetry, and evenness, make it crucial for understanding oscillatory behavior and for applications in physics and engineering.

Double Angle Identities

Double angle identities are trigonometric formulas that express trigonometric functions of a double angle (like 2?) in terms of functions of the original angle ?. These identities, such as cos(2?) = 2cos²? - 1 or cos(2?) = 1 - 2sin²?, are instrumental in simplifying expressions and solving trigonometric equations.

Trigonometric Functions

Trigonometric functions (sine, cosine, tangent, etc.) are fundamental in relating the angles of a triangle to the ratios of its sides and are extensively used in periodic phenomena, geometry, and calculus. They provide a means to model cycles, oscillations, and wave-like behavior in various fields of science and engineering.

Function Evaluation

Function evaluation is the process of substituting a specific input value into a function to calculate its corresponding output. This is a basic operation in mathematics that involves understanding how functions map inputs to outputs, which is essential for analyzing and graphing functions.

Transcript

00:01 For this question, we're asked to evaluate what g of 2 theta is.

00:06 So we know what g function equals to, equals to cosine of x.

00:11 So therefore, i can rewrite this equation as cosine of 2 theta.

00:16 And cosine of 2 theta, that's the double angle, and we have a double angle formula for cosine, and this is equal to there are three of them, so i'm just going to choose one of them, and i'm going to choose 1 minus 2, sine square of theta.

00:32 So in order to find out what sine square theta is, i need to use this figure to help me do that.

00:39 Okay, so they give you the radius of this circle because this is the equation for the circle where equals to five.

00:48 And to find the radius, you know that you should take the square root of five.

00:52 So there's your score of five.

00:54 And now, so therefore i'm looking at this angle right here.

00:58 And here's my right triangle.

01:00 So i'm going to draw a little bigger here so you can see.

01:02 It.

01:03 So here's square root of five and here's theta.

01:07 And then they also give you the point.

01:10 The x is, you know, you don't know how far it went left, but you do know how far it went up.

01:14 Y value is two...

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