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Trigonometry Lessons

Trigonometry is a word derived from Greek meaning "measurement of triangles". In its basic form it deals with the sides and angles of a triangle, hence it was used in the applications of astronomy, navigation and surveying. The other powerful perspective of trigonometry is that the trigonometric relations can be considered as functions. This page deals with basic trigonometry lessons and explains the concepts with examples. The lessons are given for the following topics.
  1. Radian Measure of an angle
  2. Right triangle trigonometry
  3. Trigonometric Identities
  4. Applications using right triangles

Trigonometry Topics

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Radian measure of an angle
You have learned the degree measure of an angle and the classification of the angles on the basis of this measure. Radians are another way of measuring an angle.

                                                                     
Definition of a Radian: One radian is the measure of the central angle θ that
intercepts an arc equal in length to the measure of the radius
'r' of the circle.
            

In general the measure of a central angle which cuts an arc of length 's', θ = $\frac{s}{r}$.
The conversion equation used between the angle and the radius measure is,
π radians = 180º.
To convert radians to degrees multiply by $\frac{180}{\pi }$ and to convert degrees to radians multiply with $\frac{\pi }{180}$
Example:
Convert 135º into radians.
We multiply by the conversion factor $\frac{\pi }{180}$
135º = $135\times \frac{\pi }{180}$ = $\frac{3\pi }{4}$ radians

Convert 2 radians to degree measures
2 radians = $2\times \frac{180}{\pi }=\frac{360}{\pi }$  ≈ 114.59º.

Six trigonometric ratios:
The six trigonometric ratios are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec) and cotangent (cot).
The six trigonometric ratios of an acute angle in a right triangle are defined in terms of the lengths of the legs and the hypotenuse as follows:

Sin A = Opposite leg / Hypotenuse = BC/AB
Cos A = Adjacent leg / Hypotenuse = CA/AB
Tan A = Opposite leg / Adjacent leg = BC/CA

Csc A = Hypotenuse / Opposite leg = AB/BC
Sec A = Hypotenuse /  Adjacent leg = AB/CA
Cot A = Adjacent leg / Opposite leg = CA/BC

                         
 
You may find the Csc, sec and cot are reciprocals of sin, cos and tangent respectively. These ratios are hence known as the reciprocal ratios.
The relationship between sin, cos and tan is
tan θ = sin θ/cos θ.

Example:
In the right triangle ABC find the six trigonometric values of angle θ.



The lengths of two legs are given as adjacent leg AB = 3 cm and opposite leg CA = 6cm
Using Pythagorean theorem, the length of the hypotenuse BC = $BC = \sqrt{AB^{2}+CA^{2}}=\sqrt{3^{2}+6^{2}}=\sqrt{45}$ ≈ 6.71 cm
Now let us calculate the six trigonometric ratios using the formulas.

Sin θ = opp leg / hypotenuse = `6/6.71`= 0.894
Cos θ = adj leg / hypotenuse = `3/6.71` = 0.447
Tan θ = opp leg / adj leg = `6/3` = 2

Now the reciprocal ratios are
Csc θ = `6.71/6` = 1.118
Sce θ = `6.71/3` = 2.237
Cot θ = `1/2`.

Trigonometric ratios of complementary angles

In the above example we calculated the trig ratios of angle B. The other acute angle in the triangle C is the complement of angle B
C = (90 - B)º.
You can verify using the above example
Sin C = Cos B = Cos (90 - C).
Cos C = Sin B = Sin (90 - B).
Tan C = Cot B = Cot (90 - B).

Sec C = Csc B = Csc (90 -C).
Csc C = Sec B = Sec (90 - C).
Cot C = Tan B = Tan (90 - C).

Trigonometric Identities
The following identities are the relationship between different trigonometric functions.
1. Reciprocal identities:
    Sin θ = 1/ Csc θ       Cos θ = 1/ Sec θ    Tan θ = 1 /Cot θ
    Csc θ = 1/ Sin θ        Sec θ = 1/ Cos θ   Cot θ = 1/ Tan θ.

2.  Quotient Identities:
     Tan θ = Sin θ / Cos θ.        Cot θ = Cos θ / Sin θ.

3. Cofunction identities
    If θ is an acute angle then the following relationships are true.
    Sin (90 - θ) = Cos θ
    Cos (90 - θ) = Sin θ
    Tan (90 - θ) = Cot θ
    Cot(90 - θ) = Tan θ
    Sec (90 - θ) = Csc θ
    Csc (90 - θ) = Sec θ

4. Pythagorean Identities:
    Sin2 θ + Cos2 θ = 1
    1 + Tan2 θ = Sec2 θ
    1 + Cot2 θ = CSC2 θ

The first two groups of identities are derived from the definitions of trigonometric ratios. Pythagorean identities get their name from the fact, that  they are proved using Pythagorean theorem for the lengths of the sides of a right triangle.

Applying trigonometric identities:
Let tan θ = 4.  Evaluate cot θ and sec θ using trigonometric identities assuming θ to be an acute angle.
cot θ = 1/ tan θ = `1/4`.         Using Reciprocal identity

sec2 θ = 1 + tan2 θ              Pythagorean identity
          = 1 + 42 = 1 + 16 = 17
sec θ = √17.                        All trig functions of an acute angle are positive.

Word problem using trigonometric ratios

In real life situations it is ofter easier to measure angles and horizontal distances rather than measuring heights and depths. The measures of heights and distances are calculated making use of the trigonometric relationship in a right triangle. The two common vocabulary used are the angle of elevation and the angle of depression.

Definitions
The angle of elevation is the angle the upward line of observation makes with the horizontal.
The angle of depression is the angle the downward line of observation makes with the horizontal.

Example:
 An observer is watching the peak of a tall monument standing at a horizontal distance of 120 ft from its base. If the angle of elevation made is
observed to be 79.2 º, find the height of the monument correct to the tenth of a feet.

        The situation is represented by the adjoining diagram.
O is the observer, and the monument is represented by the vertical line AB.
The angle of elevation is marked at observer's point.

Height AB is the opposite leg as related to the angle of elevation O.
The length of the adjacent leg OA is known.
Hence the tan ratio involving the opposite and adjacent legs can be used to
solve the problem.

Tan θ = opposite leg/adjacent leg
h/120 = tan 79.2º
h = 120 x tan 79.2º
Using calculator to evaluate the above expression
h = 120 x 5.2422
   ≈ 629.1 ft     rounded to the tenth of a ft.