91Ó°ÊÓ

(a) Use Pythagoras' Theorem in a square \(A B C D\) of side 1 to show that the diagonal \(A C\) has length \(\sqrt{2}\). Use this to work out in your head the exact values of \(\sin 45^{\circ}, \cos 45^{\circ}, \tan 45^{\circ}\) (b) In an equilateral triangle \(\triangle A B C\) with sides of length \(2,\) join \(A\) to the midpoint \(M\) of the base \(B C\). Apply Pythagoras' Theorem to find \(A M\). Hence work out in your head the exact values of \(\sin 30^{\circ}, \cos 30^{\circ}, \tan 30^{\circ},\) \(\sin 60^{\circ}, \cos 60^{\circ}, \tan 60^{\circ}\) (c)(i) On the unit circle with centre at the origin \(O:(0,0),\) mark the point \(P\) so that \(P\) lies in the first quadrant, and so that \(O P\) makes an angle \(\theta\) with the positive \(x\) -axis (measured anticlockwise from the positive \(x\) -axis). Explain why \(P\) has coordinates \((\cos \theta, \sin \theta)\). (ii) Extend the definitions of \(\cos \theta\) and \(\sin \theta\) to apply to angles beyond the first quadrant, so that for any point \(P\) on the unit circle, where \(O P\) makes an angle \(\theta\) measured anticlockwise from the positive \(x\) -axis, the coordinates of \(P\) are \((\cos \theta, \sin \theta) .\) Check that the resulting functions sin and cos satisfy: \(*\) sin and cos are both positive in the first quadrant, \(*\) sin is positive and cos is negative in the second quadrant, \(*\) sin and cos are both negative in the third quadrant, and \(*\) sin is negative and \(\cos\) is positive in the fourth quadrant. (iii) Use (a), (b) to calculate the exact values of \(\cos 315^{\circ}, \sin 225^{\circ}, \tan 210^{\circ}\), \(\cos 120^{\circ}, \sin 960^{\circ}, \tan \left(-135^{\circ}\right)\) (d) Given a circle of radius \(1,\) work out the exact area of a regular \(n\) -gon inscribed in the circle: (i) when \(n=3\) (ii) when \(n=4\) (iii) when \(n=6\) (iv) when \(n=8\) (v) when \(n=12\). (e) Given a circle of radius 1 , work out the area of a regular \(n\) -gon circumscribed around the circle: (i) when \(n=3\) (ii) when \(n=4\) (iii) when \(n=6\) (iv) when \(n=8\) (v) when \(n=12\). Knowing the exact values of \(\sin , \cos\) and tan for the special angles \(0^{\circ}\), \(30^{\circ}, 45^{\circ}, 60^{\circ}, 90^{\circ}\) is like knowing one's tables. In particular, it allows one to evaluate trigonometric functions mentally for related angles in all four quadrants (using the CAST mnemonic \(-\mathrm{C}\) being in the \(\mathrm{SE}\) of the unit circle, \(A\) in the NE quadrant, \(S\) in the NW quadrant, and \(T\) in the SW quadrant - to remind us which functions are positive in each quadrant). These special angles arise over and over again in connection with equilateral triangles, squares, regular hexagons, regular octagons, regular dodecagons, etc., where one can use what one knows to calculate exactly in geometry.

Step by step solution

01

Calculate Diagonal of Square

A square has side length 1. Using Pythagoras' Theorem, the length of the diagonal, denoted as \(AC\), can be found using: \[AC = \sqrt{(1^2 + 1^2)} = \sqrt{(2)}\] Thus, the diagonal is \(\sqrt{2}\).

02

Find Trigonometric Ratios for \(45^{\circ}\)

\(45^{\circ}\) forms an isosceles right triangle with equal legs (side length 1). Therefore: \(\sin 45^{\circ} = \cos 45^{\circ} = \frac{1}{\sqrt{2}}\), and \(\tan 45^{\circ} = \frac{1}{1} = 1\).

03

Find Midsegment of Equilateral Triangle

For an equilateral triangle \(\triangle ABC\) with side length 2, if \(M\) is the midpoint of \(BC\), \(BM = CM = 1\). The height \(AM\) can be found using Pythagoras: \(AM = \sqrt{2^2 - 1^2} = \sqrt{3}\).

04

Calculate Trigonometric Ratios for \(30^{\circ}\), \(60^{\circ}\)

Using triangle \(ABM\), \(\sin 30^{\circ} = \frac{1}{2}\), \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\), \(\tan 30^{\circ} = \frac{1}{\sqrt{3}}\). For \(60^{\circ}\), \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\), \(\cos 60^{\circ} = \frac{1}{2}\), \(\tan 60^{\circ} = \sqrt{3}\).

05

Determine Coordinates on the Unit Circle

For a point \(P\) on the unit circle, the radius (length 1) is aligned with angle \(\theta\). The coordinates \((\cos \theta, \sin \theta)\) satisfy the equation \(x^2 + y^2 = 1\).

06

Extend Trigonometric Definitions Beyond First Quadrant

Consider each quadrant: In the first \((\cos, \sin): (+,+)\), second \((-,+)\), third \((-,-)\), and fourth \((+,-)\). These reflect based on axis, maintaining positive/negative status per quadrant.

07

Find Trigonometric Values for Given Angles

Using angles derived from previously known angles: \(\cos 315^{\circ} = \cos(360^{\circ} - 45^{\circ}) = \frac{1}{\sqrt{2}}\), \(\sin 225^{\circ} = \sin(180^{\circ} + 45^{\circ}) = -\frac{1}{\sqrt{2}}\), \(\tan 210^{\circ} = \tan(180^{\circ} + 30^{\circ}) = \frac{1}{\sqrt{3}}\). Continue similarly for other angles.

08

Area of Inscribed Polygon

When \(n\) is the number of sides. For inscribed \(n\)-gon, \(\text{Area} = \frac{n}{2} R^2 \sin\left(\frac{2\pi}{n}\right)\). Calculate for \(n = 3, 4, 6, 8, 12\) with \(R = 1\).

09

Area of Circumscribed Polygon

For circumscribed \(n\)-gon, \(\text{Area} = n R^2 \tan\left(\frac{\pi}{n}\right)\). Again calculate for \(n = 3, 4, 6, 8, 12\) with \(R = 1\). Use tangent values for polynomials to achieve the result.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pythagoras' Theorem

Pythagoras' Theorem is a fundamental principle in trigonometry and geometry. It helps us find the length of a side in a right triangle when the lengths of the other two sides are known. The theorem is expressed as:\[c^2 = a^2 + b^2\]where \(c\) is the hypotenuse, the side opposite the right angle, and \(a\) and \(b\) are the other two sides.
For instance, in a square with each side measuring 1, the diagonal (hypotenuse in this context) can be determined using Pythagoras' Theorem. Since both sides are equal in length:\[AC = \sqrt{1^2 + 1^2} = \sqrt{2}\]Hence, the diagonal is \(\sqrt{2}\). This method is also applicable for finding heights or other dimensions in various geometric shapes, such as equilateral triangles, and plays a critical role in understanding trigonometric ratios for specific angles like \(45^{\circ}\). This theorem is essential for solving many problems in trigonometry, especially when analyzing or constructing right triangles.

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate system \(O:(0,0)\). It is a powerful tool in trigonometry for understanding the behavior of trigonometric functions like sine and cosine across different angles. Any point \(P\) on the unit circle has coordinates \((\cos\theta, \sin\theta)\) at an angle \(\theta\) from the positive x-axis.
The equation of the unit circle is:\[x^2 + y^2 = 1\]where \(x\) and \(y\) represent the coordinates \(\cos\theta\) and \(\sin\theta\), respectively. As \(\theta\) increases, \(P\) moves around the circle; the angle is measured in a counterclockwise direction from the x-axis. This property allows us to explore angles beyond the first quadrant, extending the definition of sine and cosine for all angles, not just those between \(0^{\circ}\) and \(90^{\circ}\).

• In the first quadrant, both sine and cosine values are positive.
• In the second quadrant, sine is positive, but cosine is negative.
• In the third quadrant, both values are negative.
• In the fourth quadrant, sine is negative, but cosine stays positive.

Understanding these concepts helps determine the exact trigonometric values for various angles using the unit circle and greatly aids in graphing and transformations.

Trigonometric Ratios

Trigonometric ratios are essential to solving various geometric problems and analyzing angles. They relate the angles of a triangle to the lengths of its sides, focusing primarily on right-angled triangles. The primary trigonometric ratios are sine, cosine, and tangent:

• Sine (\(\sin\)): the ratio of the length of the opposite side to the hypotenuse.
• Cosine (\(\cos\)): the ratio of the length of the adjacent side to the hypotenuse.
• Tangent (\(\tan\)): the ratio of the length of the opposite side to the adjacent side.

For instance, in a 45-degree right triangle (45-45-90 triangle), the legs are equal, and its trigonometric ratios are:

• \(\sin 45^{\circ} = \cos 45^{\circ} = \frac{1}{\sqrt{2}}\)
• \(\tan 45^{\circ} = 1\)

These ratios are derived using Pythagoras' Theorem and provide a basis for calculating the trigonometric values of other angles when seen in the context of similar triangles or the unit circle.

Special Angles

Special angles in trigonometry, such as \(0^{\circ}\), \(30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\), have known, exact trigonometric values that greatly simplify calculations. For example, in the context of an equilateral triangle divided into two 30-60-90 triangles, each angle yields a distinct set of trigonometric values:

• \(\sin 30^{\circ} = \frac{1}{2}\), \(\cos 30^{\circ} = \frac{\sqrt{3}}{2}\), \(\tan 30^{\circ} = \frac{1}{\sqrt{3}}\)
• \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\), \(\cos 60^{\circ} = \frac{1}{2}\), \(\tan 60^{\circ} = \sqrt{3}\)

The knowledge of these angles and their corresponding trigonometric functions allows us to solve problems more efficiently and accurately without needing calculators. They are used in various geometric configurations, such as shapes inscribed or circumscribed in circles, and provide key insights into solving trigonometric equations.