Mathematics Short TestMarks: 41QUESTION 11.1 { \frac{\tan 480^{\circ} \cdot \sin 300^{\circ} \cdot \cos 14^{\circ} \cdot \sin (-135^{\circ})}{\sin 104^{\circ} \cdot \cos 225^{\circ}} = \frac{3}{2}$} 1.2 \[ 1.2 \[ 1.2 \[ \cos 75^{\circ} =

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Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and computer science. In this article, we will focus on solving trigonometric equations, specifically the ones involving tangent, sine, and cosine functions.

Understanding the Basics

Before we dive into solving the given equations, let's review some basic trigonometric identities and formulas.

  • Tangent identity: tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}
  • Sine identity: sin⁡(θ+ϕ)=sin⁡θcos⁡ϕ+cos⁡θsin⁡ϕ\sin (\theta + \phi) = \sin \theta \cos \phi + \cos \theta \sin \phi
  • Cosine identity: cos⁡(θ+ϕ)=cos⁡θcos⁡ϕ−sin⁡θsin⁡ϕ\cos (\theta + \phi) = \cos \theta \cos \phi - \sin \theta \sin \phi

Solving the First Equation

The first equation is:

tan⁡480∘⋅sin⁡300∘⋅cos⁡14∘⋅sin⁡(−135∘)sin⁡104∘⋅cos⁡225∘=32\frac{\tan 480^{\circ} \cdot \sin 300^{\circ} \cdot \cos 14^{\circ} \cdot \sin (-135^{\circ})}{\sin 104^{\circ} \cdot \cos 225^{\circ}} = \frac{3}{2}

To solve this equation, we need to simplify the expression by using trigonometric identities and formulas.

Step 1: Simplify the Tangent Function

We can start by simplifying the tangent function using the identity tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}.

tan⁡480∘=sin⁡480∘cos⁡480∘\tan 480^{\circ} = \frac{\sin 480^{\circ}}{\cos 480^{\circ}}

Since sin⁡480∘=sin⁡(360∘+120∘)=sin⁡120∘\sin 480^{\circ} = \sin (360^{\circ} + 120^{\circ}) = \sin 120^{\circ} and cos⁡480∘=cos⁡(360∘+120∘)=−cos⁡120∘\cos 480^{\circ} = \cos (360^{\circ} + 120^{\circ}) = -\cos 120^{\circ}, we can rewrite the tangent function as:

tan⁡480∘=sin⁡120∘−cos⁡120∘\tan 480^{\circ} = \frac{\sin 120^{\circ}}{-\cos 120^{\circ}}

Step 2: Simplify the Sine Function

Next, we can simplify the sine function using the identity sin⁡(θ+ϕ)=sin⁡θcos⁡ϕ+cos⁡θsin⁡ϕ\sin (\theta + \phi) = \sin \theta \cos \phi + \cos \theta \sin \phi.

sin⁡300∘=sin⁡(180∘+120∘)=sin⁡120∘\sin 300^{\circ} = \sin (180^{\circ} + 120^{\circ}) = \sin 120^{\circ}

Step 3: Simplify the Cosine Function

We can simplify the cosine function using the identity cos⁡(θ+ϕ)=cos⁡θcos⁡ϕ−sin⁡θsin⁡ϕ\cos (\theta + \phi) = \cos \theta \cos \phi - \sin \theta \sin \phi.

cos⁡14∘=cos⁡(180∘−166∘)=cos⁡166∘\cos 14^{\circ} = \cos (180^{\circ} - 166^{\circ}) = \cos 166^{\circ}

Step 4: Simplify the Sine Function

We can simplify the sine function using the identity sin⁡(−θ)=−sin⁡θ\sin (-\theta) = -\sin \theta.

sin⁡(−135∘)=−sin⁡135∘\sin (-135^{\circ}) = -\sin 135^{\circ}

Step 5: Simplify the Sine Function

We can simplify the sine function using the identity sin⁡(θ+ϕ)=sin⁡θcos⁡ϕ+cos⁡θsin⁡ϕ\sin (\theta + \phi) = \sin \theta \cos \phi + \cos \theta \sin \phi.

sin⁡104∘=sin⁡(90∘+14∘)=sin⁡14∘\sin 104^{\circ} = \sin (90^{\circ} + 14^{\circ}) = \sin 14^{\circ}

Step 6: Simplify the Cosine Function

We can simplify the cosine function using the identity cos⁡(θ+ϕ)=cos⁡θcos⁡ϕ−sin⁡θsin⁡ϕ\cos (\theta + \phi) = \cos \theta \cos \phi - \sin \theta \sin \phi.

cos⁡225∘=cos⁡(180∘+45∘)=−cos⁡45∘\cos 225^{\circ} = \cos (180^{\circ} + 45^{\circ}) = -\cos 45^{\circ}

Step 7: Substitute the Simplified Expressions

Now, we can substitute the simplified expressions into the original equation.

sin⁡120∘−cos⁡120∘⋅sin⁡120∘⋅cos⁡166∘⋅−sin⁡135∘sin⁡14∘⋅−cos⁡45∘=32\frac{\frac{\sin 120^{\circ}}{-\cos 120^{\circ}} \cdot \sin 120^{\circ} \cdot \cos 166^{\circ} \cdot -\sin 135^{\circ}}{\sin 14^{\circ} \cdot -\cos 45^{\circ}} = \frac{3}{2}

Step 8: Simplify the Expression

We can simplify the expression by canceling out the common terms.

sin⁡120∘⋅sin⁡120∘⋅cos⁡166∘⋅sin⁡135∘−cos⁡120∘⋅sin⁡14∘⋅cos⁡45∘=32\frac{\sin 120^{\circ} \cdot \sin 120^{\circ} \cdot \cos 166^{\circ} \cdot \sin 135^{\circ}}{-\cos 120^{\circ} \cdot \sin 14^{\circ} \cdot \cos 45^{\circ}} = \frac{3}{2}

Step 9: Use Trigonometric Identities

We can use trigonometric identities to simplify the expression further.

sin⁡120∘=sin⁡(90∘+30∘)=cos⁡30∘\sin 120^{\circ} = \sin (90^{\circ} + 30^{\circ}) = \cos 30^{\circ}

sin⁡135∘=sin⁡(90∘+45∘)=cos⁡45∘\sin 135^{\circ} = \sin (90^{\circ} + 45^{\circ}) = \cos 45^{\circ}

cos⁡166∘=cos⁡(180∘−14∘)=−cos⁡14∘\cos 166^{\circ} = \cos (180^{\circ} - 14^{\circ}) = -\cos 14^{\circ}

cos⁡120∘=cos⁡(180∘−60∘)=−cos⁡60∘\cos 120^{\circ} = \cos (180^{\circ} - 60^{\circ}) = -\cos 60^{\circ}

cos⁡45∘=12\cos 45^{\circ} = \frac{1}{\sqrt{2}}

cos⁡30∘=32\cos 30^{\circ} = \frac{\sqrt{3}}{2}

cos⁡14∘=cos⁡14∘\cos 14^{\circ} = \cos 14^{\circ}

sin⁡14∘=sin⁡14∘\sin 14^{\circ} = \sin 14^{\circ}

sin⁡60∘=32\sin 60^{\circ} = \frac{\sqrt{3}}{2}

Step 10: Substitute the Simplified Expressions

Now, we can substitute the simplified expressions into the expression.

32⋅32⋅−cos⁡14∘⋅12−(−32)⋅sin⁡14∘⋅12=32\frac{\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} \cdot -\cos 14^{\circ} \cdot \frac{1}{\sqrt{2}}}{-\left(-\frac{\sqrt{3}}{2}\right) \cdot \sin 14^{\circ} \cdot \frac{1}{\sqrt{2}}} = \frac{3}{2}

Step 11: Simplify the Expression

We can simplify the expression by canceling out the common terms.

34⋅−cos⁡14∘⋅1232⋅sin⁡14∘⋅12=32\frac{\frac{3}{4} \cdot -\cos 14^{\circ} \cdot \frac{1}{\sqrt{2}}}{\frac{\sqrt{3}}{2} \cdot \sin 14^{\circ} \cdot \frac{1}{\sqrt{2}}} = \frac{3}{2}

Step 12: Use Trigonometric Identities

We can use trigonometric identities to simplify the expression further.

34⋅−cos⁡14∘⋅12=34⋅−cos⁡14∘⋅12\frac{3}{4} \cdot -\cos 14^{\circ} \cdot \frac{1}{\sqrt{2}} = \frac{3}{4} \cdot -\cos 14^{\circ} \cdot \frac{1}{\sqrt{2}}

32⋅sin⁡14∘⋅12=32⋅sin⁡14∘⋅12\frac{\sqrt{3}}{2} \cdot \sin 14^{\circ} \cdot \frac{1}{\sqrt{2}} = \frac{\sqrt{3}}{2} \cdot \sin 14^{\circ} \cdot \frac{1}{\sqrt{2}}

Step 13: Simplify the Expression

We can simplify the expression by canceling out the common terms.

−34⋅cos⁡14∘32⋅sin⁡14∘=32\frac{-\frac{3}{4} \cdot \cos 14^{\circ}}{\frac{\sqrt{3}}{2} \cdot \sin 14^{\circ}} = \frac{3}{2}

Step 14: Use Trigonometric Identities

We can use trigonometric identities to simplify the expression further.

−34⋅cos⁡14∘32⋅sin⁡14∘=−34⋅cos⁡14∘32⋅sin⁡14∘\frac{-\frac{3}{4} \cdot \cos 14^{\circ}}{\frac{\sqrt{3}}{2} \cdot \sin 14^{\circ}} = \frac{-\frac{3}{4} \cdot \cos 14^{\circ}}{\frac{\sqrt{3}}{2} \cdot \sin 14^{\circ}}

Step 15: Simplify the Expression

We can simplify the expression by canceling out the common terms.

−34⋅cos⁡14∘32⋅sin⁡14∘=−34⋅cos⁡14∘32⋅sin⁡14∘\frac{-\frac{3}{4} \cdot \cos 14^{\circ}}{\frac{\sqrt{3}}{2} \cdot \sin 14^{\circ}} = \frac{-\frac{3}{4} \cdot \cos 14^{\circ}}{\frac{\sqrt{3}}{2} \cdot \sin 14^{\circ}}

Step 16: Use Trigonometric Identities

We can use trigonometric identities to simplify the expression further.

Q&A Section

Q: What is the main difference between the sine, cosine, and tangent functions?

A: The sine, cosine, and tangent functions are all trigonometric functions that describe the relationships between the sides and angles of triangles. The main difference between them is the ratio of the sides they represent. The sine function represents the ratio of the opposite side to the hypotenuse, the cosine function represents the ratio of the adjacent side to the hypotenuse, and the tangent function represents the ratio of the opposite side to the adjacent side.

Q: How do I simplify a trigonometric expression using trigonometric identities?

A: To simplify a trigonometric expression using trigonometric identities, you need to identify the relevant identities and apply them to the expression. For example, if you have an expression involving the sine and cosine functions, you can use the identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1 to simplify it.

Q: What is the difference between the sine and cosine functions?

A: The sine and cosine functions are both trigonometric functions that describe the relationships between the sides and angles of triangles. The main difference between them is the ratio of the sides they represent. The sine function represents the ratio of the opposite side to the hypotenuse, while the cosine function represents the ratio of the adjacent side to the hypotenuse.

Q: How do I solve a trigonometric equation involving the tangent function?

A: To solve a trigonometric equation involving the tangent function, you need to isolate the tangent function and then use the identity tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta} to simplify it. You can then use algebraic techniques to solve for the angle.

Q: What is the difference between the sine and cosine functions in terms of their periodicity?

A: The sine and cosine functions are both periodic functions, meaning they repeat themselves at regular intervals. However, the sine function has a period of 2π2\pi, while the cosine function has a period of 2π2\pi as well. This means that both functions repeat themselves every 2π2\pi radians.

Q: How do I simplify a trigonometric expression involving multiple angles?

A: To simplify a trigonometric expression involving multiple angles, you need to use trigonometric identities to combine the angles. For example, if you have an expression involving the sine and cosine functions of two angles, you can use the identity sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin (A + B) = \sin A \cos B + \cos A \sin B to simplify it.

Q: What is the difference between the sine and cosine functions in terms of their range?

A: The sine and cosine functions both have a range of [−1,1][-1, 1], meaning they can take on any value between -1 and 1. However, the sine function is an odd function, meaning it is symmetric about the origin, while the cosine function is an even function, meaning it is symmetric about the y-axis.

Q: How do I solve a trigonometric equation involving the sine function?

A: To solve a trigonometric equation involving the sine function, you need to isolate the sine function and then use the identity sin⁡θ=sin⁡(A+B)\sin \theta = \sin (A + B) to simplify it. You can then use algebraic techniques to solve for the angle.

Q: What is the difference between the sine and cosine functions in terms of their derivatives?

A: The sine and cosine functions both have derivatives that are equal to their negative counterparts. Specifically, the derivative of the sine function is the cosine function, and the derivative of the cosine function is the negative sine function.

Q: How do I simplify a trigonometric expression involving complex numbers?

A: To simplify a trigonometric expression involving complex numbers, you need to use the properties of complex numbers to simplify the expression. For example, if you have an expression involving the sine and cosine functions of a complex number, you can use the identity sin⁡(z)=sin⁡(x+iy)=sin⁡xcos⁡iy+cos⁡xsin⁡iy\sin (z) = \sin (x + iy) = \sin x \cos iy + \cos x \sin iy to simplify it.

Q: What is the difference between the sine and cosine functions in terms of their integrals?

A: The sine and cosine functions both have integrals that are equal to their negative counterparts. Specifically, the integral of the sine function is the negative cosine function, and the integral of the cosine function is the negative sine function.

Q: How do I solve a trigonometric equation involving the cosine function?

A: To solve a trigonometric equation involving the cosine function, you need to isolate the cosine function and then use the identity cos⁡θ=cos⁡(A+B)\cos \theta = \cos (A + B) to simplify it. You can then use algebraic techniques to solve for the angle.

Q: What is the difference between the sine and cosine functions in terms of their applications?

A: The sine and cosine functions both have numerous applications in various fields, including physics, engineering, and computer science. However, the sine function is often used to describe periodic phenomena, such as sound waves and light waves, while the cosine function is often used to describe rotational phenomena, such as the motion of a pendulum.