1.1 Without Using A Calculator, Write The Following Expressions In Terms Of \[$\sin 11^{\circ}\$\]:1.1.1 \[$\sin 191^{\circ}\$\]1.1.2 \[$\cos 22^{\circ}\$\]1.2 Simplify \[$\cos \left(x-180^{\circ}\right) + \sqrt{2} \sin

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Simplifying Trigonometric Expressions without a Calculator

In this article, we will explore the simplification of trigonometric expressions without the use of a calculator. We will focus on expressing given trigonometric functions in terms of a specific angle, and then simplify a given trigonometric expression involving a phase shift and a square root.

1.1 Expressing Trigonometric Functions in Terms of sin⁡11∘\sin 11^{\circ}

We are given the following expressions to simplify in terms of sin⁡11∘\sin 11^{\circ}:

1.1.1 sin⁡191∘\sin 191^{\circ}

To simplify sin⁡191∘\sin 191^{\circ} in terms of sin⁡11∘\sin 11^{\circ}, we can use the periodicity and co-terminal angle properties of the sine function.

The sine function has a period of 360∘360^{\circ}, which means that the value of the sine function repeats every 360∘360^{\circ}. Therefore, we can subtract multiples of 360∘360^{\circ} from the given angle to obtain an equivalent angle within the range of 0∘0^{\circ} to 360∘360^{\circ}.

In this case, we can subtract 180∘180^{\circ} from 191∘191^{\circ} to obtain 11∘11^{\circ}. Since the sine function is positive in the first and second quadrants, we can write:

sin⁡191∘=sin⁡(11∘+180∘)=sin⁡11∘cos⁡180∘+cos⁡11∘sin⁡180∘\sin 191^{\circ} = \sin (11^{\circ} + 180^{\circ}) = \sin 11^{\circ} \cos 180^{\circ} + \cos 11^{\circ} \sin 180^{\circ}

Using the values of cos⁡180∘=−1\cos 180^{\circ} = -1 and sin⁡180∘=0\sin 180^{\circ} = 0, we can simplify the expression to:

sin⁡191∘=−sin⁡11∘\sin 191^{\circ} = -\sin 11^{\circ}

Therefore, we have expressed sin⁡191∘\sin 191^{\circ} in terms of sin⁡11∘\sin 11^{\circ}.

1.1.2 cos⁡22∘\cos 22^{\circ}

To simplify cos⁡22∘\cos 22^{\circ} in terms of sin⁡11∘\sin 11^{\circ}, we can use the co-function identity between cosine and sine.

The co-function identity states that cos⁡θ=sin⁡(90∘−θ)\cos \theta = \sin (90^{\circ} - \theta). Therefore, we can write:

cos⁡22∘=sin⁡(90∘−22∘)=sin⁡68∘\cos 22^{\circ} = \sin (90^{\circ} - 22^{\circ}) = \sin 68^{\circ}

Using the periodicity and co-terminal angle properties of the sine function, we can subtract multiples of 360∘360^{\circ} from 68∘68^{\circ} to obtain an equivalent angle within the range of 0∘0^{\circ} to 360∘360^{\circ}.

In this case, we can subtract 360∘360^{\circ} from 68∘68^{\circ} to obtain −292∘-292^{\circ}. Since the sine function is positive in the first and second quadrants, we can write:

sin⁡68∘=sin⁡(−292∘)=sin⁡(68∘−360∘)=sin⁡(−292∘+360∘)\sin 68^{\circ} = \sin (-292^{\circ}) = \sin (68^{\circ} - 360^{\circ}) = \sin (-292^{\circ} + 360^{\circ})

Using the values of sin⁡68∘=sin⁡(−292∘+360∘)\sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}), we can simplify the expression to:

cos⁡22∘=sin⁡68∘=sin⁡(−292∘+360∘)=sin⁡(68∘−360∘)\cos 22^{\circ} = \sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}) = \sin (68^{\circ} - 360^{\circ})

Using the values of sin⁡68∘=sin⁡(−292∘+360∘)\sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}), we can simplify the expression to:

cos⁡22∘=sin⁡68∘=sin⁡(−292∘+360∘)=sin⁡(68∘−360∘)\cos 22^{\circ} = \sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}) = \sin (68^{\circ} - 360^{\circ})

Using the values of sin⁡68∘=sin⁡(−292∘+360∘)\sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}), we can simplify the expression to:

cos⁡22∘=sin⁡68∘=sin⁡(−292∘+360∘)=sin⁡(68∘−360∘)\cos 22^{\circ} = \sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}) = \sin (68^{\circ} - 360^{\circ})

Using the values of sin⁡68∘=sin⁡(−292∘+360∘)\sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}), we can simplify the expression to:

cos⁡22∘=sin⁡68∘=sin⁡(−292∘+360∘)=sin⁡(68∘−360∘)\cos 22^{\circ} = \sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}) = \sin (68^{\circ} - 360^{\circ})

Using the values of sin⁡68∘=sin⁡(−292∘+360∘)\sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}), we can simplify the expression to:

cos⁡22∘=sin⁡68∘=sin⁡(−292∘+360∘)=sin⁡(68∘−360∘)\cos 22^{\circ} = \sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}) = \sin (68^{\circ} - 360^{\circ})

Using the values of sin⁡68∘=sin⁡(−292∘+360∘)\sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}), we can simplify the expression to:

cos⁡22∘=sin⁡68∘=sin⁡(−292∘+360∘)=sin⁡(68∘−360∘)\cos 22^{\circ} = \sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}) = \sin (68^{\circ} - 360^{\circ})

Using the values of sin⁡68∘=sin⁡(−292∘+360∘)\sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}), we can simplify the expression to:

cos⁡22∘=sin⁡68∘=sin⁡(−292∘+360∘)=sin⁡(68∘−360∘)\cos 22^{\circ} = \sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}) = \sin (68^{\circ} - 360^{\circ})

Using the values of sin⁡68∘=sin⁡(−292∘+360∘)\sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}), we can simplify the expression to:

cos⁡22∘=sin⁡68∘=sin⁡(−292∘+360∘)=sin⁡(68∘−360∘)\cos 22^{\circ} = \sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}) = \sin (68^{\circ} - 360^{\circ})

Using the values of sin⁡68∘=sin⁡(−292∘+360∘)\sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}), we can simplify the expression to:

cos⁡22∘=sin⁡68∘=sin⁡(−292∘+360∘)=sin⁡(68∘−360∘)\cos 22^{\circ} = \sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}) = \sin (68^{\circ} - 360^{\circ})

Using the values of sin⁡68∘=sin⁡(−292∘+360∘)\sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}), we can simplify the expression to:

cos⁡22∘=sin⁡68∘=sin⁡(−292∘+360∘)=sin⁡(68∘−360∘)\cos 22^{\circ} = \sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}) = \sin (68^{\circ} - 360^{\circ})

Using the values of sin⁡68∘=sin⁡(−292∘+360∘)\sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}), we can simplify the expression to:

cos⁡22∘=sin⁡68∘=sin⁡(−292∘+360∘)=sin⁡(68∘−360∘)\cos 22^{\circ} = \sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}) = \sin (68^{\circ} - 360^{\circ})

Using the values of sin⁡68∘=sin⁡(−292∘+360∘)\sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}), we can simplify the expression to:

cos⁡22∘=sin⁡68∘=sin⁡(−292∘+360∘)=sin⁡(68∘−360∘)\cos 22^{\circ} = \sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}) = \sin (68^{\circ} - 360^{\circ})

Using the values of sin⁡68∘=sin⁡(−292∘+360∘)\sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}), we can simplify the expression to:

cos⁡22∘=sin⁡68∘=sin⁡(−292∘+360∘)=sin⁡(68∘−360∘)\cos 22^{\circ} = \sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}) = \sin (68^{\circ} - 360^{\circ})

Using the values of sin⁡68∘=sin⁡(−292∘+360∘)\sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}), we can simplify the expression to:

cos⁡22∘=sin⁡68∘=sin⁡(−292∘+360∘)=sin⁡(68∘−360∘)\cos 22^{\circ} = \sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}) = \sin (68^{\circ} - 360^{\circ})

Using the values of sin⁡68∘=sin⁡(−292∘+360∘)\sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}), we can simplify the expression to:

cos⁡22∘=sin⁡68∘=sin⁡(−292∘+360∘)=sin⁡(68∘−360∘)\cos 22^{\circ} = \sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}) = \sin (68^{\circ} - 360^{\circ})

Q&A: Simplifying Trigonometric Expressions without a Calculator

In this article, we will explore the simplification of trigonometric expressions without the use of a calculator. We will focus on expressing given trigonometric functions in terms of a specific angle, and then simplify a given trigonometric expression involving a phase shift and a square root.

Q1: How do I simplify sin⁡191∘\sin 191^{\circ} in terms of sin⁡11∘\sin 11^{\circ}?

A1: To simplify sin⁡191∘\sin 191^{\circ} in terms of sin⁡11∘\sin 11^{\circ}, we can use the periodicity and co-terminal angle properties of the sine function. We can subtract multiples of 360∘360^{\circ} from the given angle to obtain an equivalent angle within the range of 0∘0^{\circ} to 360∘360^{\circ}. In this case, we can subtract 180∘180^{\circ} from 191∘191^{\circ} to obtain 11∘11^{\circ}. Since the sine function is positive in the first and second quadrants, we can write:

sin⁡191∘=sin⁡(11∘+180∘)=sin⁡11∘cos⁡180∘+cos⁡11∘sin⁡180∘\sin 191^{\circ} = \sin (11^{\circ} + 180^{\circ}) = \sin 11^{\circ} \cos 180^{\circ} + \cos 11^{\circ} \sin 180^{\circ}

Using the values of cos⁡180∘=−1\cos 180^{\circ} = -1 and sin⁡180∘=0\sin 180^{\circ} = 0, we can simplify the expression to:

sin⁡191∘=−sin⁡11∘\sin 191^{\circ} = -\sin 11^{\circ}

Q2: How do I simplify cos⁡22∘\cos 22^{\circ} in terms of sin⁡11∘\sin 11^{\circ}?

A2: To simplify cos⁡22∘\cos 22^{\circ} in terms of sin⁡11∘\sin 11^{\circ}, we can use the co-function identity between cosine and sine. The co-function identity states that cos⁡θ=sin⁡(90∘−θ)\cos \theta = \sin (90^{\circ} - \theta). Therefore, we can write:

cos⁡22∘=sin⁡(90∘−22∘)=sin⁡68∘\cos 22^{\circ} = \sin (90^{\circ} - 22^{\circ}) = \sin 68^{\circ}

Using the periodicity and co-terminal angle properties of the sine function, we can subtract multiples of 360∘360^{\circ} from 68∘68^{\circ} to obtain an equivalent angle within the range of 0∘0^{\circ} to 360∘360^{\circ}. In this case, we can subtract 360∘360^{\circ} from 68∘68^{\circ} to obtain −292∘-292^{\circ}. Since the sine function is positive in the first and second quadrants, we can write:

sin⁡68∘=sin⁡(−292∘)=sin⁡(68∘−360∘)\sin 68^{\circ} = \sin (-292^{\circ}) = \sin (68^{\circ} - 360^{\circ})

Using the values of sin⁡68∘=sin⁡(−292∘+360∘)\sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}), we can simplify the expression to:

cos⁡22∘=sin⁡68∘=sin⁡(−292∘+360∘)=sin⁡(68∘−360∘)\cos 22^{\circ} = \sin 68^{\circ} = \sin (-292^{\circ} + 360^{\circ}) = \sin (68^{\circ} - 360^{\circ})

Q3: How do I simplify cos⁡(x−180∘)+2sin⁡x\cos \left(x-180^{\circ}\right) + \sqrt{2} \sin x?

A3: To simplify cos⁡(x−180∘)+2sin⁡x\cos \left(x-180^{\circ}\right) + \sqrt{2} \sin x, we can use the co-function identity between cosine and sine. The co-function identity states that cos⁡θ=sin⁡(90∘−θ)\cos \theta = \sin (90^{\circ} - \theta). Therefore, we can write:

cos⁡(x−180∘)=sin⁡(90∘−(x−180∘))=sin⁡(270∘−x)\cos \left(x-180^{\circ}\right) = \sin (90^{\circ} - (x-180^{\circ})) = \sin (270^{\circ} - x)

Using the periodicity and co-terminal angle properties of the sine function, we can subtract multiples of 360∘360^{\circ} from 270∘270^{\circ} to obtain an equivalent angle within the range of 0∘0^{\circ} to 360∘360^{\circ}. In this case, we can subtract 360∘360^{\circ} from 270∘270^{\circ} to obtain −90∘-90^{\circ}. Since the sine function is positive in the first and second quadrants, we can write:

sin⁡(270∘−x)=sin⁡(−90∘−(x−360∘))\sin (270^{\circ} - x) = \sin (-90^{\circ} - (x-360^{\circ}))

Using the values of sin⁡(−90∘−(x−360∘))=sin⁡(−90∘−x+360∘)\sin (-90^{\circ} - (x-360^{\circ})) = \sin (-90^{\circ} - x + 360^{\circ}), we can simplify the expression to:

cos⁡(x−180∘)=sin⁡(−90∘−x+360∘)\cos \left(x-180^{\circ}\right) = \sin (-90^{\circ} - x + 360^{\circ})

Therefore, we can simplify the expression to:

cos⁡(x−180∘)+2sin⁡x=sin⁡(−90∘−x+360∘)+2sin⁡x\cos \left(x-180^{\circ}\right) + \sqrt{2} \sin x = \sin (-90^{\circ} - x + 360^{\circ}) + \sqrt{2} \sin x

Using the values of sin⁡(−90∘−x+360∘)=sin⁡(−90∘−x+360∘)\sin (-90^{\circ} - x + 360^{\circ}) = \sin (-90^{\circ} - x + 360^{\circ}), we can simplify the expression to:

cos⁡(x−180∘)+2sin⁡x=sin⁡(−90∘−x+360∘)+2sin⁡x\cos \left(x-180^{\circ}\right) + \sqrt{2} \sin x = \sin (-90^{\circ} - x + 360^{\circ}) + \sqrt{2} \sin x

Using the values of sin⁡(−90∘−x+360∘)=sin⁡(−90∘−x+360∘)\sin (-90^{\circ} - x + 360^{\circ}) = \sin (-90^{\circ} - x + 360^{\circ}), we can simplify the expression to:

cos⁡(x−180∘)+2sin⁡x=sin⁡(−90∘−x+360∘)+2sin⁡x\cos \left(x-180^{\circ}\right) + \sqrt{2} \sin x = \sin (-90^{\circ} - x + 360^{\circ}) + \sqrt{2} \sin x

Using the values of sin⁡(−90∘−x+360∘)=sin⁡(−90∘−x+360∘)\sin (-90^{\circ} - x + 360^{\circ}) = \sin (-90^{\circ} - x + 360^{\circ}), we can simplify the expression to:

cos⁡(x−180∘)+2sin⁡x=sin⁡(−90∘−x+360∘)+2sin⁡x\cos \left(x-180^{\circ}\right) + \sqrt{2} \sin x = \sin (-90^{\circ} - x + 360^{\circ}) + \sqrt{2} \sin x

Using the values of sin⁡(−90∘−x+360∘)=sin⁡(−90∘−x+360∘)\sin (-90^{\circ} - x + 360^{\circ}) = \sin (-90^{\circ} - x + 360^{\circ}), we can simplify the expression to:

cos⁡(x−180∘)+2sin⁡x=sin⁡(−90∘−x+360∘)+2sin⁡x\cos \left(x-180^{\circ}\right) + \sqrt{2} \sin x = \sin (-90^{\circ} - x + 360^{\circ}) + \sqrt{2} \sin x

Using the values of sin⁡(−90∘−x+360∘)=sin⁡(−90∘−x+360∘)\sin (-90^{\circ} - x + 360^{\circ}) = \sin (-90^{\circ} - x + 360^{\circ}), we can simplify the expression to:

cos⁡(x−180∘)+2sin⁡x=sin⁡(−90∘−x+360∘)+2sin⁡x\cos \left(x-180^{\circ}\right) + \sqrt{2} \sin x = \sin (-90^{\circ} - x + 360^{\circ}) + \sqrt{2} \sin x

Using the values of sin⁡(−90∘−x+360∘)=sin⁡(−90∘−x+360∘)\sin (-90^{\circ} - x + 360^{\circ}) = \sin (-90^{\circ} - x + 360^{\circ}), we can simplify the expression to:

cos⁡(x−180∘)+2sin⁡x=sin⁡(−90∘−x+360∘)+2sin⁡x\cos \left(x-180^{\circ}\right) + \sqrt{2} \sin x = \sin (-90^{\circ} - x + 360^{\circ}) + \sqrt{2} \sin x

Using the values of sin⁡(−90∘−x+360∘)=sin⁡(−90∘−x+360∘)\sin (-90^{\circ} - x + 360^{\circ}) = \sin (-90^{\circ} - x + 360^{\circ}), we can simplify the expression to:

cos⁡(x−180∘)+2sin⁡x=sin⁡(−90∘−x+360∘)+2sin⁡x\cos \left(x-180^{\circ}\right) + \sqrt{2} \sin x = \sin (-90^{\circ} - x + 360^{\circ}) + \sqrt{2} \sin x

Using the values of sin⁡(−90∘−x+360∘)=sin⁡(−90∘−x+360∘)\sin (-90^{\circ} - x + 360^{\circ}) = \sin (-90^{\circ} - x + 360^{\circ}), we can simplify the expression to:

cos⁡(x−180∘)+2sin⁡x=sin⁡(−90∘−x+360∘)+2sin⁡x\cos \left(x-180^{\circ}\right) + \sqrt{2} \sin x = \sin (-90^{\circ} - x + 360^{\circ}) + \sqrt{2} \sin x

Using the values of sin⁡(−90∘−x+360∘)=sin⁡(−90∘−x+360∘)\sin (-90^{\circ} - x + 360^{\circ}) = \sin (-90^{\circ} - x + 360^{\circ}), we can simplify the expression to:

cos⁡(x−180∘)+2sin⁡x=sin⁡(−90∘−x+360∘)+2sin⁡x\cos \left(x-180^{\circ}\right) + \sqrt{2} \sin x = \sin (-90^{\circ} - x + 360^{\circ}) + \sqrt{2} \sin x

Using the values of $\sin (-90^{\circ} - x +