📏 trigonometry

1. **Problem statement:** Calculate the value of $$\sin^2 35^\circ + \sin^2 10^\circ + \sqrt{2} \sin 35^\circ \sin 10^\circ$$. 2. **Formula and rules:** Recall the identity for sin

1. **State the problem:** Verify the trigonometric identities and solve the given trigonometric questions. 2. **Identity verification:**

1. **Problem statement:** Find the angle $\theta$ for point A(-3,4) in the domain $0 < \theta \leq 4\pi$. 2. **Formula and rules:** The angle $\theta$ in standard position is found

1. The problem is to clarify whether counting angles from $C270$ starts at 0 or 1. 2. In angle measurement, counting always starts at 0 degrees (or radians), representing the initi

1. The problem is about understanding how to count angles when dealing with transformations of the sine function, specifically for the function $-2\sin x$ and counting from $270^\c

1. Problem. Identify the transformations applied to the parent sine function to obtain $f(x) = -2\sin x$.

1. Problem. Identify the transformations applied to the parent sine function to obtain $f(x) = -2\sin x$.

1. **State the problem:** Solve the equation $$4 \sin \theta \cos \theta + \cos^2 \theta = 2 - \sin \theta$$ for $$0^\circ \leq \theta \leq 360^\circ$$. 2. **Recall identities and

1. The problem is to understand and work with a trigonometric function. 2. Trigonometric functions relate angles to ratios of sides in right triangles and are periodic functions.

1. **Problem statement:** We have two right triangles sharing an angle of 15°. The top-right triangle has hypotenuse $r=75$ and height $x$. We want to find $x$. 2. **Formula used:*

1. The problem is to verify the identity: $$\sin^4 x - \cos^4 x = 1 - 2\cos^2 x$$. 2. Recall the difference of squares formula: $$a^2 - b^2 = (a-b)(a+b)$$.

1. **State the problem:** Prove or verify the identity $$(\sin A + \cos A)^2 + (\sin A - \cos A)^2 = 2.$$\n\n2. **Recall the formula:** The square of a sum and difference are given

1. **Problem statement:** Solve triangle ABC given $A=43^\circ$, $b=7$ cm, and $c=6$ cm. 2. **Known values:**

1. Let's understand where the number 0.707 comes from. 2. The value 0.707 is approximately equal to $\frac{1}{\sqrt{2}}$.

1. We are asked to show that $$\frac{\cot^2 \alpha}{\csc \alpha - 1} = \csc \alpha + 1$$. 2. Recall the identities:

1. Diberikan bahwa $\sin 23^\circ = m$. Kita diminta mencari nilai $\cos 113^\circ$.\n\n2. Gunakan identitas sudut pelengkap dan hubungan antara sinus dan kosinus: $$\cos \theta =

1. The problem is to find the value of $\tan 135^\circ$. 2. Recall the tangent function and its properties: $\tan(\theta) = \frac{\sin \theta}{\cos \theta}$.

1. **State the problem:** We want to show that $$3 \tan^2 \theta + 5 \sin^2 \theta \equiv \frac{8 \sin^2 \theta - 5 \sin^4 \theta}{1 - \sin^2 \theta}$$ for any angle $$\theta$$. 2.

1. **State the problem:** An observer sees two planes at different altitudes and angles of elevation. We need to find the distance between the two planes. 2. **Given:**

1. **Problem:** Solve the equation $\sin(x + \frac{\pi}{6}) + \sin(x - \frac{\pi}{6}) = \frac{1}{2}$ for $x$ in $[0, 2\pi]$. 2. **Formula and rules:** Use the sum-to-product identi

1. **State the problem:** Simplify the expression $$\frac{\cos(\theta)}{1 - \sin(\theta)} - \tan(\theta)$$. 2. **Recall formulas and identities:**