Find the acute angle solution to the following equation involving cofunctions. θ \theta θ is in degrees. cos ( 2 θ + 15 ) = sin ( 5 θ + 12 ) \cos\left(2\theta+15\right)=\sin\left(5\theta+12\right) cos ( 2 θ + 15 ) = sin ( 5 θ + 12 )

Let's see if we can solve this problem. So here we're asked to find the acute angle solution to the following equation involving cofunctions. And we're given that theta is in degrees. Now what I notice about this equation is that we have a cosine operation on the left side and a sine operation on the right side. This means that there's not really a direct way we can solve this right off the bat, so we're going to have to use co functions in this situation. Now what I'm gonna do for this problem is I'm gonna keep the left side the same. So we'll have the cosine of 2 theta plus 15 on the left, And then this is going to equal the right side of the equation, which I'm gonna change the sine to a cosine. And I can do this because the cosine is the cofunction of sine. But in order to complete this, I need to put the complementary angle in here rather than what we have. And the complementary angle is just going to be 90 degrees minus our original angle, which would be the expression up here, 5 theta plus 12. Now from here what I'm going to do is I'm going to take this negative sign, I'm going to distribute it into the parenthesis. So on the left side, we'll continue to have this cosine of 2 theta plus 15. And then on the right side, we're going to have the cosine of 90, and that's gonna be minus 5 theta minus 12. Now at this point, what I can do is I can combine this 90 and this negative 12, because 90 minus 12 is going to give you 78. So I have 78 minus 5 theta on the right side, And then on the left side, we're going to have cosine of 2 theta plus 15. Notice how this left side has not changed since we haven't done anything to it. But notice at this point, we have the same operation on both sides, we have this cosine. So that means what we can do is set the insides of the equations equal. So So we're going to have 2 theta plus 15, and that whole thing is going to equal 78 minus 5 theta. So now that we have the n sides equal to each other, what we can do is solve for theta in this equation. Now the first step is going to be to subtract this 15 on both sides of the equation. That'll get the 15s to cancel on the left side. So we'll end up with 2 theta is equal to, and then 78 minus 15. Now 78 minus 10 would give you 68, and you subtract 5 from that, you'll get 63. And that's gonna be minus 5 theta. Now at this point, what I can do is add this 5 theta on both sides of the equation. That's going to get the 5 thetas to cancel on the right side, leaving us with just 63. And then on the left side, we'll have 5 theta plus 2 theta, which is equal to 7 theta. Now from here, what I can do is I can solve for theta by dividing 7 on both sides of the equation. That's going to get the sevens to cancel on the left side, leaving us with just theta. And then 63 divided by 7 is 9. So our angle theta is equal to 9 degrees, and that is the solution to this problem. So hope you found this video helpful. Thanks for watching, and please let me know if you have any questions.

8. Trigonometric Functions on Right Triangles

Cofunctions of Complementary Angles

Precalculus

8. Trigonometric Functions on Right Triangles

Cofunctions of Complementary Angles

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Multiple Choice

Find the acute angle solution to the following equation involving cofunctions. $\theta$ is in degrees.
$\cos\left(2\theta+15\right)=\sin\left(5\theta+12\right)$

$1\degree$

$4\degree$

$6\degree$

$9\degree$

Verified step by step guidance

Recognize that the equation involves cofunctions: $ \cos(2\theta + 15) = \sin(5\theta + 12) $. Cofunctions are related by the identity $ \cos(\theta) = \sin(90^\circ - \theta) $.

Apply the cofunction identity to rewrite the equation: $ \cos(2\theta + 15) = \sin(90^\circ - (2\theta + 15)) $.

Set the expressions inside the sine function equal to each other: $ 90^\circ - (2\theta + 15) = 5\theta + 12 $.

Simplify the equation: $ 90^\circ - 2\theta - 15 = 5\theta + 12 $. Combine like terms to form a linear equation in terms of $ \theta $.

Solve the linear equation for $ \theta $ to find the acute angle solution. Ensure $ \theta $ is within the range of acute angles, which is between $ 0^\circ $ and $ 90^\circ $.