What is the Value of sinA if tanA secA 3?

This guide provides a step-by-step solution to the equation tanA secA 3 to determine the value of sinA. We will use trigonometric identities and algebraic manipulation to find the answer. This method will be explained in detail below.

Introduction to Trigonometric Identities and the Given Equation

In trigonometry, the relationship between sine, cosine, and tangent functions is crucial for solving various mathematical problems. The given equation is tanA secA 3. To find the value of sinA, we need to express the given equation in terms of sine and cosine functions using known trigonometric identities.

Expressing tanA and secA in Terms of sinA and cosA

First, recall the definitions of tangent and secant functions in terms of sine and cosine:

tanA frac{sinA}{cosA} secA frac{1}{cosA}

Substituting these definitions into the given equation, we get:

(frac{sinA}{cosA} cdot frac{1}{cosA} 3)

Combining and Simplifying the Equation

Multiplying both sides by cosA ( eq 0)), we have:

(frac{sinA cdot 1}{cosA cdot cosA} 3)

Combining the terms on the left-hand side, we get:

(frac{sinA cdot 1}{cos^2 A} 3)

Which simplifies to:

(frac{sinA}{cosA} cdot frac{1}{cosA} 3)

Multiplying both sides by cosA) again:

(sinA cdot 1 3 cdot cosA)

Expressing sinA in Terms of cosA

From the above equation, we can express sinA as:

(sinA 3 cdot cosA - 1)

Next, we use the Pythagorean identity sin^2 A cos^2 A 1) to further simplify:

((3 cdot cosA - 1)^2 cos^2 A 1)

Further Algebraic Manipulation Using the Pythagorean Identity

Expanding the left-hand side:

(9 cdot cos^2 A - 6 cdot cosA 1 cos^2 A 1)

Combining like terms:

(10 cdot cos^2 A - 6 cdot cosA 1 1)

Subtracting 1 from both sides:

(10 cdot cos^2 A - 6 cdot cosA 0)

Factoring out cosA):

(cosA(10 cdot cosA - 6) 0)

This gives us two cases:

cosA 0) (which is not valid since it would make secA undefined) 10 cdot cosA - 6 0)

For the second case:

10 cdot cosA 6 quad Rightarrow quad cosA frac{3}{5})

Finding sinA Using the Pythagorean Identity

Now, we use the Pythagorean identity to find sinA:

((frac{3}{5})^2 sin^2 A 1)

Calculating:

(frac{9}{25} sin^2 A 1)

Subtracting (frac{9}{25}) from both sides:

(sin^2 A 1 - frac{9}{25} frac{16}{25})

Taking the square root:

(sinA pm frac{4}{5})

Thus, the value of sinA is:

(sinA frac{4}{5}) or (sinA -frac{4}{5})

The answer depends on the quadrant in which angle A lies.

Alternative Method Using Half-Angle Formulas and Identities

An alternative method involves using half-angle formulas and identities. Starting with the given equation:

(frac{sinA}{cosA} cdot frac{1}{cosA} 3)

We can express sinA as:

(sinA frac{1}{2}(frac{tanA}{2} frac{1}{tanA/2}))

Using the value of tanA/2 from solving the equation, we get:

(sinA frac{1}{2} cdot frac{2tanA/2}{1 - tan^2A/2})

Given that tanA/2 frac{1}{2}), substituting gives:

(sinA frac{1}{2} cdot frac{2 cdot frac{1}{2}}{1 - (frac{1}{2})^2})

Simplifying:

(sinA frac{1}{2} cdot frac{1}{1 - frac{1}{4}} frac{1}{2} cdot frac{1}{frac{3}{4}} frac{2}{3})

Which simplifies to:

(sinA frac{4}{5})

Verification Using Pythagorean Theorem with a Right Triangle

Finally, we can verify the solution using a right triangle. Given secA frac{3}{5}), we can deduce the sides of the triangle:

Adjacent side (P) 3 Hypotenuse (H) 5 Opposite side (B) sqrt{5^2 - 3^2} sqrt{25 - 9} sqrt{16} 4

Thus, the ratio is:

(sinA frac{opposite}{hypotenuse} frac{4}{5})

This verifies our solution using the right triangle method.