Trigonometry MCQ Quiz in हिन्दी - Objective Question with Answer for Trigonometry - मुफ्त [PDF] डाउनलोड करें

In trigonometry, the sine of an angle is equivalent to the cosine of its complementary angle. Given $\cos ({75}^{\circ })=0.2588$, then $\sin ({15}^{\circ })=\cos ({75}^{\circ })=0.2588$. Therefore, the correct answer is option 3. Option 1 is the cosine of 15 degrees, option 2 is incorrect as it is the sine of 30 degrees, and option 4 is incorrect as it is the sine of 30 degrees.

The cosine of an angle is equal to the sine of its complementary angle. Since 37 and 53 are complementary angles (they sum to 90 degrees), $\cos ({53}^{\circ })=\sin ({37}^{\circ })=0.6018$. Therefore, the correct answer is option 1. Option 2 is the cosine of 37 degrees, option 3 is the sine of 45 degrees, and option 4 is the cosine of 30 degrees, none of which are applicable here.

In a right triangle, for angles that are equal, such as 45 degrees, the sine and cosine values are the same. Given $\cos ({45}^{\circ })=\frac{\sqrt{2}}{2}$, $\sin ({45}^{\circ })$ is also $\frac{\sqrt{2}}{2}$. The correct answer is option 3. Option 1 is the sine of 30 degrees, option 2 is the sine of 60 degrees, and option 4 is the sine of 90 degrees, which are incorrect for this scenario.

The correct answer is $\frac{7}{50}$. It's given that triangle ABC is similar to triangle DEF, where angle A corresponds to angle D and angle C corresponds to angle F. In similar triangles, the tangents of corresponding angles are equal. Since angle A and angle D are corresponding angles, if $\tan (A)=\frac{50}{7}$, then $\tan (D)=\frac{50}{7}$. It's also given that angles C and F are right angles. It follows that triangle DEF is a right triangle with acute angles D and E. The tangent of one acute angle in a right triangle is the inverse of the tangent of the other acute angle in the triangle. Therefore, $\tan (E)=\frac{1}{\tan (D)}$. Substituting $\frac{50}{7}$ for tan(D) in this equation yields $\tan (E)=\frac{1}{\frac{50}{7}}$, or $\tan (E)=\frac{7}{50}$. Thus, if $\tan (A)=\frac{50}{7}$, the value of tan(E) is $\frac{7}{50}$. Note that 7/50 and .14 are examples of ways to enter a correct answer.

The correct answer is $\frac{21}{29}$. It's given that triangle ABC is similar to triangle DEF, where angle A corresponds to angle D and angles C and F are right angles. In similar triangles, the tangents of corresponding angles are equal. Therefore, if $\tan A=\frac{21}{20}$, then $\tan D=\frac{21}{20}$. In a right triangle, the tangent of an acute angle is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. Therefore, in triangle DEF, if $\tan D=\frac{21}{20}$, the ratio of the length of $\overline{EF}$ to the length of $\overline{DF}$ is $\frac{21}{20}$. If the lengths of $\overline{EF}$ and $\overline{DF}$ are 21 and 20, respectively, then the ratio of the length of $\overline{EF}$ to the length of $\frac{20}{DF}$ is $\frac{21}{20}$. In a right triangle, the sine of an acute angle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse. Therefore, the value of sin D is the ratio of the length of $\overline{EF}$ to the length of $\overline{DE}$. The length of $\overline{DE}$ can be calculated using the Pythagorean theorem, which states that if the lengths of the legs of a right triangle are a and b and the length of the hypotenuse is c, then a² + b² = c². Therefore, if the lengths of $\overline{EF}$ and $\overline{DF}$ are 21 and 20, respectively, then (21)² + (20)² = (DE)², or 841 = (DE)². Taking the positive square root of both sides of this equation yields 29 = DE. Therefore, if the lengths of $\overline{EF}$ and $\overline{DF}$ are 21 and 20, respectively, then the length of $\overline{DE}$ is 29 and the ratio of the length of $\overline{EF}$ to the length of $\overline{DE}$ is $\frac{21}{29}$. Thus, if $\tan A=\frac{21}{20}$, the value of sin D is $\frac{21}{29}$. Note that 21/29, .7241, and 0.724 are examples of ways to enter a correct answer.

Choice D is correct. Corresponding angles in similar triangles have equal measures. It's given that triangle ABC is similar to triangle DEF, where A corresponds to D, so the measure of angle A is equal to the measure of angle D. Therefore, if $\tan (A)=\sqrt{3}$, then $\tan (D)=\sqrt{3}$. It's given that angles C and F are right angles, so triangles ABC and DEF are right triangles. The adjacent side of an acute angle in a right triangle is the side closest to the angle that is not the hypotenuse. It follows that the adjacent side of angle D is side DF. The opposite side of an acute angle in a right triangle is the side across from the acute angle. It follows that the opposite side of angle D is side EF. The tangent of an acute angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. Therefore, $\tan (D)=\frac{EF}{DF}$. If DF = 125, the length of side EF can be found by substituting $\sqrt{3}$ for tan(D) and 125 for DF in the equation $\tan (D)=\frac{EF}{DF}$, which yields $\sqrt{3}=\frac{EF}{125}$. Multiplying both sides of this equation by 125 yields $125\sqrt{3}=EF$. Since the length of side DF is $\sqrt{3}$ times the length of side DF, it follows that triangle DEF is a special right triangle with angle measures 30°, 60°, and 90°. Therefore, the length of the hypotenuse, DE, is 2 times the length of side DF, or DE = 2(DF). Substituting 125 for DF in this equation yields DE = 2(125), or DE = 250. Thus, if $\tan (A)=\sqrt{3}$ and DF = 125, the length of DE is 250.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the length of EF, not DE.