Given Tangent theta = -5 on 4 and cos theta >0, find sine theta and sec theta
not familiar with the expession "-5 on 4"
do you mean -5/4 ?
if tanØ = -5/4 and cosØ >0
then we know Ø is in quadrant IV
by Pythagoras, the hypotenuse of the standard triangle is √41
sinØ = -5/√41
sec Ø = √41/4
To find the values of sine theta and sec theta, we will use the given information that:
Tangent theta = -5/4 (negative value) and cos theta > 0 (positive value).
Step 1: Find the value of sine theta.
Since tangent theta is defined as the ratio of sine theta to cosine theta, we can use the given information to find sine theta. Recall that:
Tangent theta = sine theta / cos theta
-5/4 = sine theta / cos theta
Step 2: Determine the signs of sine theta and cos theta.
Since tangent theta is negative (-5/4), one of the values of sine theta and cos theta must be positive while the other is negative. However, the given condition states that cos theta is positive. Therefore, sine theta must be negative.
Step 3: Find sine theta.
Since sine theta is negative, we need to find the negative value that satisfies the tangent theta equation. Let's assume that sine theta is -k (where k is a positive value):
-5/4 = (-k) / (cos theta)
Cross-multiplying, we get:
-5 * cos theta = -4k
Dividing by -5, we find:
cos theta = 4/5k
Since cos theta is positive, k must also be positive. Taking the reciprocal of both sides, we get:
sec theta = 5k/4
Therefore, the value of sec theta is 5k/4, where k is a positive value.
To summarize the answer:
- Sine theta = -k, where k is a positive value.
- Sec theta = 5k/4, where k is a positive value.
To find the values of sine theta and sec theta, we need to use the trigonometric identities and information given.
First, we are given that tangent theta is -5 on 4. We can interpret this as:
tangent theta = opposite/adjacent = -5/4
Next, we are told that cos theta is greater than 0. Since cos theta is the ratio of the adjacent side to the hypotenuse, a positive value indicates that the adjacent side is positive (or not negative), while the hypotenuse can be positive or negative.
Now, let's solve for the missing side using the Pythagorean Theorem:
opposite^2 + adjacent^2 = hypotenuse^2
(-5)^2 + 4^2 = hypotenuse^2
25 + 16 = hypotenuse^2
41 = hypotenuse^2
Taking the square root of both sides, we find:
sqrt(41) = hypotenuse
Since the adjacent side is positive and the hypotenuse can be either positive or negative, we can conclude that theta is in the third quadrant, where all trigonometric functions are negative except for tangent.
Now, we can find the values of sine theta and sec theta:
sine theta = opposite/hypotenuse = (-5)/sqrt(41), where sine theta is negative in the third quadrant.
sec theta = 1/cos theta = 1/(adjacent/hypotenuse) = sqrt(41)/4
Therefore, sine theta = (-5)/sqrt(41) and sec theta = sqrt(41)/4.