Question

The value of $\theta$ with $0\le\theta\le90^{\circ}$ and $\sin^{2}\theta+2\cos^{2}\theta=\frac{7}{4}$ is equal to

• A. $15^{\circ}$
• B. $30^{\circ}$
• C. $45^{\circ}$
• D. $60^{\circ}$
• E. $75^{\circ}$

Hint:

Trigonometry Tip: You can also split $2\cos^2\theta$ into $\cos^2\theta + \cos^2\theta$. Then $(\sin^2\theta + \cos^2\theta) + \cos^2\theta = 1 + \cos^2\theta = 7/4$, which makes the simplification almost instant!

Answer 1

The Correct Option is B

Concept:
To solve an equation containing multiple trigonometric functions (like both sine and cosine), use fundamental identities to convert the entire equation into a single function. The Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ is ideal for transforming squared terms.

Step 1: State the given equation.
We are given the trigonometric equation: $$\sin^2\theta + 2\cos^2\theta = \frac{7}{4}$$

Step 2: Apply the Pythagorean identity.
Recall that $\sin^2\theta = 1 - \cos^2\theta$. Substitute this expression into the given equation to entirely eliminate the sine term: $$(1 - \cos^2\theta) + 2\cos^2\theta = \frac{7}{4}$$

Step 3: Simplify the equation.
Combine the like $\cos^2\theta$ terms algebraically: $$1 + \cos^2\theta = \frac{7}{4}$$

Step 4: Isolate the trigonometric function.
Subtract 1 (which is $\frac{4}{4}$) from both sides to solve for $\cos^2\theta$: $$\cos^2\theta = \frac{7}{4} - \frac{4}{4}$$ $$\cos^2\theta = \frac{3}{4}$$ Take the square root of both sides. Since $0 \le \theta \le 90^\circ$ (first quadrant), cosine must be positive: $$\cos\theta = \frac{\sqrt{3}}{2}$$

Step 5: Determine the angle $\theta$.
Recall the standard unit circle values. The angle in the first quadrant whose cosine is $\frac{\sqrt{3}}{2}$ is 30 degrees. $$\theta = 30^\circ$$ Hence the correct answer is (B) $30^{\circ$}.