Mid-Chapter Quiz
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The diagonal divides the rectangle into two right triangles. The missing length of a right triangle can be found using the Pythagorean Theorem.
Substitute the trigonometric ratios into the given identity.
9 inches
See solution.
We are told that the front of a computer monitor is usually measured along the diagonal of the screen. We want to find the height h of the monitor.
Notice that the shape of the computer monitor is rectangular. The diagonal divides the rectangle into two right triangles. Let's consider one of the triangles.
Because h is a leg of a right triangle, we can use the Pythagorean Theorem.
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Pythagorean Theorem |
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For a right triangle, the length of the hypotenuse squared equals the sum of the squares of the lengths of the legs, or a^2+b^2=c^2. |
In our case, a is the side measuring 12, b is the leg whose length is h, and c is the hypotenuse, which measures 15. Let's substitute the values of a, b, and c into the equation and solve for h.
Substitute values
Calculate power
LHS-144=RHS-144
sqrt(LHS)=sqrt(RHS)
The height h is 9 inches.
We want to show that cot θ= cos θsin θ using the diagram. We know that the diagonal of the computer monitor divides it into two right triangles. Let's look once again at one of these triangles.
The expression that we want to verify involves trigonometric ratios. Let's recall their definitions!
| Name | Definition | Notation |
|---|---|---|
| Sine of ∠θ | Length of side opposite ∠θ/Hypotenuse | sin θ=opp/hyp |
| Cosine of ∠θ | Length of side adjacent to ∠θ/Hypotenuse | cos θ=adj/hyp |
| Tangent of ∠θ | Length of side opposite ∠θ/Length of side adjacent to ∠θ | tan θ=opp/adj |
| Cotangent of ∠θ | Length of side adjacent to ∠θ/Length of side opposite ∠θ | cot θ=adj/opp |
Let's substitute these definitions in the identity that we want to verify. cot θ= cos θsin θ ⇔ adj/opp= adj hyp/opp hyp Now, we can pick any of the acute angles and mark the relative positions of the sides to one of the acute angles.
When considering the angle that measures θ, the length of the side adjacent ( adj) to the angle is 12, the hypotenuse ( hyp) measures 15, while the length of the side opposite ( opp) to the angle is 9. Now let's substitute these values in the equation and simplify.
Substitute values
a/b=a * 18/b * 18
a/18* 18 = a
Since the resulting statement is true, we have verified that cot θ= cos θsin θ using the measures of the right triangle.