# Solve Situational Problems Involving Trigonometric Identities

• MHB
• ukumure
In summary, the conversation discusses solving situational problems involving trigonometric identities, specifically finding the values of $\sin\theta$ and $\cos\theta$ using the Pythagorean theorem. The final result is $\cos{\theta} + \sin{\theta} = \dfrac{4}{\sqrt{106}}$.
ukumure
Hi! I am so confused about the given and what is being asked, I don't know how to solve it. This topic is solving situational problems involving trigonometric identities. Your help would be a big one for me :) Thank you so much in advance!

First, we need to establish $\sin\theta$ and $\cos\theta$.
$9^2+(-5)^2=106$ (Pythagorean theorem)
so $\sin\theta$ is $\sqrt{\frac{|-5|}{106}}, \text{that is}, \left(\frac{opp}{hyp}\right)$ and $\cos\theta$ is $\frac{3}{\sqrt{106}}, \text{that is}, \left(\frac{adj}{hyp}\right)$ (recall that $\sin\theta$ is the magnitude of the opposite side of the right-angled triangle containing $\theta$ divided by the hypotenuse)

Hence $\sin\theta+\cos\theta=\frac{3+\sqrt{|-5|}}{\sqrt{106}}$.

Greg said:
First, we need to establish $\sin\theta$ and $\cos\theta$.
$9^2+(-5)^2=106$ (Pythagorean theorem)
so $\sin\theta$ is $\sqrt{\frac{|-5|}{106}}, \text{that is}, \left(\frac{opp}{hyp}\right)$ and $\cos\theta$ is $\frac{3}{\sqrt{106}}, \text{that is}, \left(\frac{adj}{hyp}\right)$ (recall that $\sin\theta$ is the magnitude of the opposite side of the right-angled triangle containing $\theta$ divided by the hypotenuse)

Hence $\sin\theta+\cos\theta=\frac{3+\sqrt{|-5|}}{\sqrt{106}}$.
THANK YOU SO MUCH! )

$\cos{\theta} = \dfrac{x}{r} = \dfrac{9}{\sqrt{106}}$

$\sin{\theta} = \dfrac{y}{r} = \dfrac{-5}{\sqrt{106}}$

$\cos{\theta} + \sin{\theta} = \dfrac{4}{\sqrt{106}}$

## 1. What are trigonometric identities?

Trigonometric identities are equations that involve trigonometric functions (such as sine, cosine, and tangent) and are true for all values of the variables in the equation. They are used to simplify and solve equations involving trigonometric functions.

## 2. How do I solve situational problems involving trigonometric identities?

To solve situational problems involving trigonometric identities, you need to first identify the given information and the unknown quantity. Then, use the given information to manipulate the equation using trigonometric identities until you are left with the unknown quantity on one side of the equation. Finally, use algebraic techniques to solve for the unknown quantity.

## 3. What are some common trigonometric identities?

Some common trigonometric identities include the Pythagorean identities (such as sin²θ + cos²θ = 1), the double angle identities (such as sin2θ = 2sinθcosθ), and the sum and difference identities (such as sin(α ± β) = sinαcosβ ± cosαsinβ).

## 4. How can I remember all the trigonometric identities?

It can be helpful to memorize the common trigonometric identities, but it is also important to understand how they are derived. You can use mnemonic devices or create your own study aids to help you remember them. Practice and repetition can also aid in memorization.

## 5. Are there any tips for solving situational problems involving trigonometric identities?

Some tips for solving situational problems involving trigonometric identities include: clearly labeling the given information and the unknown quantity, using diagrams or visual aids to help visualize the problem, and checking your solution by plugging it back into the original equation. It is also important to be familiar with the properties and rules of trigonometric functions.

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