Find the exact value of each expression.
(a) tan(sec^−1 5)
(b) sin(2sin^-1(5/13))
for these, always draw the right triangle!
for example, sin(arctan 17/8) = 15/17
Draw the triangle, fill in the missing side, and take the needed trig function. Then, as in the second problem here, remember your double-angle formula:
sin2θ = 2sinθcosθ
To find the exact value of each expression, we will need to use trigonometric identities and properties.
(a) tan(sec^−1 5):
Let's start by using the definition of sec^−1(x). Recall that sec^−1(x) is the inverse of sec(x), which means it gives us the angle whose secant is equal to x.
So, sec(sec^−1 5) = 5.
Now, we can use the definition of tan(x) to find the exact value:
tan(sec^−1 5) = sin(sec^−1 5) / cos(sec^−1 5).
Using the Pythagorean identity, we know that sin^2(x) + cos^2(x) = 1.
Since sec(x) = 5, we can rewrite this as:
1 + tan^2(x) = sec^2(x).
Substituting 5 for sec(x), we get:
1 + tan^2(sec^−1 5) = 5^2.
Simplifying, we have:
1 + tan^2(sec^−1 5) = 25.
tan^2(sec^−1 5) = 25 - 1.
tan^2(sec^−1 5) = 24.
Taking the square root of both sides, we get:
tan(sec^−1 5) = ±√24.
Since tan(x) is positive in the first and third quadrant, we can take the positive square root:
tan(sec^−1 5) = √24.
(b) sin(2sin^-1(5/13)):
Using the double angle formula for sine, we have:
sin(2θ) = 2sin(θ)cos(θ).
In this case, θ = sin^−1(5/13).
We can determine the values of sin(θ) and cos(θ) by using a right-angled triangle. Since sin(θ) = opposite/hypotenuse and cos(θ) = adjacent/hypotenuse, let's create a right triangle with opposite side length 5 and hypotenuse 13.
The adjacent side can be found using the Pythagorean theorem:
adjacent^2 + 5^2 = 13^2.
adjacent^2 + 25 = 169.
adjacent^2 = 169 - 25.
adjacent^2 = 144.
adjacent = √144.
adjacent = 12.
Now, we can find sin(θ) and cos(θ) using the ratios:
sin(θ) = 5/13.
cos(θ) = 12/13.
Plugging these values into the double angle formula, we have:
sin(2sin^−1(5/13)) = 2(sin(sin^−1(5/13)))cos(sin^−1(5/13)).
Using the fact that sin(sin^−1(x)) = x and cos(sin^−1(x)) = √(1 - x^2), we can simplify this expression:
sin(2sin^−1(5/13)) = 2(5/13)(12/13).
sin(2sin^−1(5/13)) = (10/13)(12/13).
sin(2sin^−1(5/13)) = 120/169.
Therefore, the exact value of each expression is:
(a) tan(sec^−1 5) = √24.
(b) sin(2sin^−1(5/13)) = 120/169.
To find the exact value of each expression, we'll need to use the properties of trigonometric functions and inverse trigonometric functions. Let's start with the first expression.
(a) tan(sec^−1 5):
1. Start by identifying the innermost function, which is sec^−1 5. This means finding an angle whose secant is equal to 5.
2. Recall that secant is the reciprocal of cosine, so sec^−1 5 is the angle whose cosine is 1/5. Let's call this angle θ.
3. Use the Pythagorean identity: cos^2 θ + sin^2 θ = 1, to find the value of sin θ.
Since cos θ = 1/5, square it: (1/5)^2 = 1/25
Substitute this value into the identity: (1/25) + sin^2 θ = 1
Solve for sin θ: sin^2 θ = 1 - 1/25 = 24/25
Take the positive square root since we're dealing with a positive angle: sin θ = √(24/25) = √24/5
4. Now, we need to find the value of tan(sec^−1 5). Recall that tangent is the ratio of sine and cosine, so tan θ = sin θ / cos θ.
Substitute the values: tan(sec^−1 5) = (√24/5) / (1/5) = (√24/5) * (5/1) = √24
Therefore, tan(sec^−1 5) = √24.
Now let's move on to the second expression.
(b) sin(2sin^−1(5/13)):
1. Begin by identifying the innermost function, which is sin^−1 (5/13). This means finding an angle whose sine is equal to 5/13.
2. Let's call this angle φ. We have sin φ = 5/13.
3. To find the value of sin(2sin^−1(5/13)), we'll use the double-angle formula for sine: sin(2x) = 2sin(x)cos(x).
4. Substitute sin φ into the formula: sin(2sin^−1(5/13)) = 2(sin sin^−1(5/13))(cos sin^−1(5/13)).
5. Use the Pythagorean identity: sin^2 x + cos^2 x = 1, to find the value of cos sin^−1(5/13).
Since sin sin^−1(5/13) = 5/13, square it: (5/13)^2 = 25/169
Substitute this value into the identity: sin^2 sin^−1(5/13) + cos^2 sin^−1(5/13) = 1
Solve for cos sin^−1(5/13): cos^2 sin^−1(5/13) = 1 - 25/169 = 144/169
Take the positive square root since we're dealing with a positive angle: cos sin^−1(5/13) = √(144/169) = 12/13
6. Substitute the values: sin(2sin^−1(5/13)) = 2(sin sin^−1(5/13))(cos sin^−1(5/13)) = 2(5/13)(12/13) = 120/169
Therefore, sin(2sin^−1(5/13)) = 120/169.
So, the exact values of the expressions are:
(a) tan(sec^−1 5) = √24
(b) sin(2sin^−1(5/13)) = 120/169