Trigonometric Identities for Solving for Exact Values

In summary, the conversation is discussing how to find the exact values for sine, cosine, and tangent of an angle A, given that sine A = 3/5 and the angle is between 90 and 180 degrees. The conversation explains that the Pythagorean Identity can be used to find the values for cosine, and then the Double-Angle Formulae can be used to find the values for sine, cosine, and tangent.
  • #1
pavadrin
156
0
hey i have a problem which I am puzzled over. i I am given that:
[tex]\sin A = \frac{3}{5}[/tex]
and i am asked to find exact values for:
[tex]\sin 2A[/tex]
[tex]\cos 2A[/tex]
[tex]\tan 2A[/tex]
where [tex]90^0 \leq A \leq 180^0[/tex] (the power to zero is the degree sign)
i have gone about solving this by the use of a pythagorien (?) triple, therefore
[tex]\cos A = \frac{4}{5}[/tex] and
[tex]\tan A = \frac{3}{4}[/tex].
however i do not understand how A can be greator than 90 since this is in a right angle triangle. thanks in advance for those who help,
Pavadrin.
 
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  • #2
pavadrin said:
i have gone about solving this by the use of a pythagorien (?) triple, therefore
[tex]\cos A = \frac{4}{5}[/tex] and
[tex]\tan A = \frac{3}{4}[/tex].
however i do not understand how A can be greator than 90 since this is in a right angle triangle. thanks in advance for those who help,
Pavadrin.
The Pythagorean Identity states that: sin2x + cos2x = 1, where x is some angle.
So in this case, we have:
sin2A + cos2A = 1
[tex]\Leftrightarrow \cos ^ 2 A = 1 - \sin ^ 2 A = 1 - \left( \frac{3}{5} \right) ^ 2 = \frac{16}{25}[/tex]
So there will be 2 possible value for cosA:
[tex]\Leftrightarrow \cos A = \pm \frac{4}{5}[/tex], right?
Since (4 / 5)2, and (-4 / 5)2 both return 16 / 25.
So how can we know what the value of cos A is? The problem gives us more information, that the angle A is between 90o, and 180o, i.e in the second quadrant. By looking at the unit circle, can you see what sign cos A takes? Is it positive or negative?
After having cos A, one can use the Double-Angle Formulae to finish the problem:
sin(2A) = 2 sin(A) cos(A)
cos(2A) = cos2(A) - sin2(A) = 1 - 2sin2(A) = 2cos2(A) - 1.
Ok, can you go from here? :)
 
  • #3
hmmmm...i see. i think i understand that now, thanks,
Pavadrin
 

What are the basic trigonometry identities?

The basic trigonometry identities are sine, cosine, tangent, cosecant, secant, and cotangent. They are used to relate the ratios of the sides of a right triangle to its angles.

What is the Pythagorean identity in trigonometry?

The Pythagorean identity in trigonometry is sin²x + cos²x = 1. It states that the square of the sine of an angle plus the square of the cosine of the same angle is always equal to 1.

How are double angle identities used in trigonometry?

Double angle identities are used to express trigonometric functions of double angles in terms of trigonometric functions of a single angle. This is useful for simplifying complex trigonometric expressions and solving equations.

What are the reciprocal identities in trigonometry?

The reciprocal identities in trigonometry are cosecant (csc), secant (sec), and cotangent (cot). They are the inverse or reciprocal of sine, cosine, and tangent, respectively.

Why are trigonometry identities important in science?

Trigonometry identities are important in science because they are used in a variety of fields such as physics, engineering, and astronomy to calculate and understand the relationships between angles and sides in triangles. They also help in solving complex mathematical equations and modeling real-world phenomena.

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