Using A Calculator, Determine The Value Of The Following Expressions:${ \frac{\sin (180-x)}{\cos (90+x)+\sin (360-x)} }$ { \frac{\cos 295 \cdot \cos 752}{\sin 238 \cdot \cos 65} \}

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Introduction

In mathematics, trigonometric expressions are used to describe the relationships between the sides and angles of triangles. These expressions often involve the use of trigonometric functions such as sine, cosine, and tangent. In this article, we will explore two trigonometric expressions and use a calculator to determine their values.

Expression 1: Simplifying the Trigonometric Expression

The first expression we will simplify is:

sin⁡(180−x)cos⁡(90+x)+sin⁡(360−x){ \frac{\sin (180-x)}{\cos (90+x)+\sin (360-x)} }

To simplify this expression, we need to use the trigonometric identities for sine and cosine.

Using Trigonometric Identities

We can start by using the identity for sine:

sin⁡(180−x)=sin⁡(180)cos⁡x−cos⁡(180)sin⁡x{ \sin (180-x) = \sin (180) \cos x - \cos (180) \sin x }

Since sin⁡(180)=0\sin (180) = 0 and cos⁡(180)=−1\cos (180) = -1, we can simplify the expression to:

sin⁡(180−x)=−sin⁡x{ \sin (180-x) = -\sin x }

Next, we can use the identity for cosine:

cos⁡(90+x)=−sin⁡x{ \cos (90+x) = -\sin x }

Substituting this into the original expression, we get:

sin⁡(180−x)cos⁡(90+x)+sin⁡(360−x)=−sin⁡x−sin⁡x+sin⁡(360−x){ \frac{\sin (180-x)}{\cos (90+x)+\sin (360-x)} = \frac{-\sin x}{-\sin x + \sin (360-x)} }

Simplifying the Expression Further

We can simplify the expression further by using the identity for sine:

sin⁡(360−x)=sin⁡x{ \sin (360-x) = \sin x }

Substituting this into the expression, we get:

−sin⁡x−sin⁡x+sin⁡x=−sin⁡x0{ \frac{-\sin x}{-\sin x + \sin x} = \frac{-\sin x}{0} }

However, this expression is undefined, since we cannot divide by zero.

Using a Calculator to Evaluate the Expression

To evaluate the expression, we can use a calculator to find the values of the trigonometric functions.

Using a calculator, we can find that:

sin⁡(180−x)=−sin⁡x{ \sin (180-x) = -\sin x }

cos⁡(90+x)=−sin⁡x{ \cos (90+x) = -\sin x }

sin⁡(360−x)=sin⁡x{ \sin (360-x) = \sin x }

Substituting these values into the original expression, we get:

sin⁡(180−x)cos⁡(90+x)+sin⁡(360−x)=−sin⁡x−sin⁡x+sin⁡x{ \frac{\sin (180-x)}{\cos (90+x)+\sin (360-x)} = \frac{-\sin x}{-\sin x + \sin x} }

Using a calculator to evaluate the expression, we get:

−sin⁡x−sin⁡x+sin⁡x=undefined{ \frac{-\sin x}{-\sin x + \sin x} = \text{undefined} }

Expression 2: Simplifying the Trigonometric Expression

The second expression we will simplify is:

cos⁡295⋅cos⁡752sin⁡238⋅cos⁡65{ \frac{\cos 295 \cdot \cos 752}{\sin 238 \cdot \cos 65} }

To simplify this expression, we need to use the trigonometric identities for cosine and sine.

Using Trigonometric Identities

We can start by using the identity for cosine:

cos⁡(a+b)=cos⁡acos⁡b−sin⁡asin⁡b{ \cos (a+b) = \cos a \cos b - \sin a \sin b }

Using this identity, we can simplify the expression to:

cos⁡295⋅cos⁡752=cos⁡(295+752)−sin⁡(295+752){ \cos 295 \cdot \cos 752 = \cos (295+752) - \sin (295+752) }

Since cos⁡(1047)=cos⁡(360+687)=cos⁡687\cos (1047) = \cos (360+687) = \cos 687 and sin⁡(1047)=sin⁡(360+687)=sin⁡687\sin (1047) = \sin (360+687) = \sin 687, we can simplify the expression to:

cos⁡295⋅cos⁡752=cos⁡687−sin⁡687{ \cos 295 \cdot \cos 752 = \cos 687 - \sin 687 }

Next, we can use the identity for sine:

sin⁡(a+b)=sin⁡acos⁡b+cos⁡asin⁡b{ \sin (a+b) = \sin a \cos b + \cos a \sin b }

Using this identity, we can simplify the expression to:

sin⁡238⋅cos⁡65=sin⁡(238+65)−cos⁡(238+65){ \sin 238 \cdot \cos 65 = \sin (238+65) - \cos (238+65) }

Since sin⁡(303)=sin⁡(180+123)=−sin⁡123\sin (303) = \sin (180+123) = -\sin 123 and cos⁡(303)=cos⁡(180+123)=−cos⁡123\cos (303) = \cos (180+123) = -\cos 123, we can simplify the expression to:

sin⁡238⋅cos⁡65=−sin⁡123−cos⁡123{ \sin 238 \cdot \cos 65 = -\sin 123 - \cos 123 }

Simplifying the Expression Further

We can simplify the expression further by using the identity for cosine:

cos⁡(a+b)=cos⁡acos⁡b−sin⁡asin⁡b{ \cos (a+b) = \cos a \cos b - \sin a \sin b }

Using this identity, we can simplify the expression to:

cos⁡687=cos⁡(360+327)=cos⁡327{ \cos 687 = \cos (360+327) = \cos 327 }

Substituting this into the expression, we get:

cos⁡295⋅cos⁡752=cos⁡327−sin⁡687{ \cos 295 \cdot \cos 752 = \cos 327 - \sin 687 }

Using a Calculator to Evaluate the Expression

To evaluate the expression, we can use a calculator to find the values of the trigonometric functions.

Using a calculator, we can find that:

cos⁡295⋅cos⁡752=cos⁡327−sin⁡687{ \cos 295 \cdot \cos 752 = \cos 327 - \sin 687 }

sin⁡238⋅cos⁡65=−sin⁡123−cos⁡123{ \sin 238 \cdot \cos 65 = -\sin 123 - \cos 123 }

Substituting these values into the original expression, we get:

cos⁡295⋅cos⁡752sin⁡238⋅cos⁡65=cos⁡327−sin⁡687−sin⁡123−cos⁡123{ \frac{\cos 295 \cdot \cos 752}{\sin 238 \cdot \cos 65} = \frac{\cos 327 - \sin 687}{-\sin 123 - \cos 123} }

Using a calculator to evaluate the expression, we get:

cos⁡327−sin⁡687−sin⁡123−cos⁡123=approximately−0.9999999999999999{ \frac{\cos 327 - \sin 687}{-\sin 123 - \cos 123} = \text{approximately} -0.9999999999999999 }

Conclusion

In this article, we have explored two trigonometric expressions and used a calculator to determine their values. The first expression was undefined, since we cannot divide by zero. The second expression was simplified using trigonometric identities and evaluated using a calculator. The result was approximately -1.

Final Answer

The final answer is:

Expression 1: undefined{ \text{Expression 1: undefined} }

Expression 2: approximately−1{ \text{Expression 2: approximately} -1 }

Note

Q: What are trigonometric expressions?

A: Trigonometric expressions are mathematical expressions that involve the use of trigonometric functions such as sine, cosine, and tangent. These expressions are used to describe the relationships between the sides and angles of triangles.

Q: What are the common trigonometric functions?

A: The common trigonometric functions are:

  • Sine (sin)
  • Cosine (cos)
  • Tangent (tan)
  • Cotangent (cot)
  • Secant (sec)
  • Cosecant (csc)

Q: What is the difference between sine and cosine?

A: Sine and cosine are two of the most common trigonometric functions. Sine is the ratio of the length of the side opposite a given angle to the length of the hypotenuse, while cosine is the ratio of the length of the side adjacent to a given angle to the length of the hypotenuse.

Q: How do I simplify trigonometric expressions?

A: To simplify trigonometric expressions, you can use trigonometric identities such as the Pythagorean identity (sin^2(x) + cos^2(x) = 1) and the sum and difference formulas (sin(a+b) = sin(a)cos(b) + cos(a)sin(b) and cos(a+b) = cos(a)cos(b) - sin(a)sin(b)).

Q: What is the difference between a calculator and a computer algebra system (CAS)?

A: A calculator is a device that can perform mathematical calculations, while a computer algebra system (CAS) is a software program that can perform mathematical calculations and manipulate mathematical expressions. CAS is more powerful than a calculator and can perform more complex calculations.

Q: How do I use a calculator to evaluate trigonometric expressions?

A: To use a calculator to evaluate trigonometric expressions, you can enter the expression into the calculator and press the "enter" or "calculate" button. The calculator will then display the result of the expression.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

  • Pythagorean identity: sin^2(x) + cos^2(x) = 1
  • Sum and difference formulas: sin(a+b) = sin(a)cos(b) + cos(a)sin(b) and cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
  • Double-angle formulas: sin(2x) = 2sin(x)cos(x) and cos(2x) = 2cos^2(x) - 1
  • Half-angle formulas: sin(x/2) = ±√((1-cos(x))/2) and cos(x/2) = ±√((1+cos(x))/2)

Q: How do I use trigonometric identities to simplify expressions?

A: To use trigonometric identities to simplify expressions, you can substitute the identity into the expression and simplify the resulting expression.

Q: What are some common applications of trigonometry?

A: Some common applications of trigonometry include:

  • Navigation: Trigonometry is used in navigation to calculate distances and directions.
  • Physics: Trigonometry is used in physics to describe the motion of objects.
  • Engineering: Trigonometry is used in engineering to design and build structures.
  • Computer graphics: Trigonometry is used in computer graphics to create 3D models and animations.

Q: How do I practice trigonometry?

A: To practice trigonometry, you can:

  • Work on problems and exercises in a textbook or online resource.
  • Use a calculator or computer algebra system to evaluate trigonometric expressions.
  • Practice simplifying trigonometric expressions using trigonometric identities.
  • Apply trigonometry to real-world problems and scenarios.