Question
Find the trigonometric functions of : 45°
✓
Answer
Angle of measure $45^{\circ}$ :
Let $m \angle X O A=45^{\circ}$
Its terminal arm (ray $O A$ ) intersects the standard unit circle at $P(x, y)$. Draw seg PM perpendicular to the $X$-axis.
$\therefore \triangle \mathrm{OMP}$ is a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle.
$ \mathrm{OP} =1,$
$\mathrm{OM} =\frac{1}{\sqrt{2}} \mathrm{OP}$
$ =\frac{1}{\sqrt{2}}(1)$
$ =\frac{1}{\sqrt{2}}$
$\mathrm{PM} =\frac{1}{\sqrt{2}} \mathrm{OP}$
$ =\frac{1}{\sqrt{2}}(1)$
$ =\frac{1}{\sqrt{2}} $
Since point $P$ lies in the 1 st quadrant, $x>0, y>0$
$\therefore \mathrm{x}=\mathrm{OM}=\frac{1}{\sqrt{2}}$ and
$ y=P M=\frac{1}{\sqrt{2}}$
$\therefore P=\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right) $
$\therefore \mathrm{P}=\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$
$\sin 45^{\circ}=y=\frac{1}{\sqrt{2}}$
$\cos 45^{\circ}=x=\frac{1}{\sqrt{2}}$
$\tan 45^{\circ}=\frac{y}{x}=\frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}}=1$
$\operatorname{cosec} 45^{\circ}=\frac{1}{y}=\frac{1}{\left(\frac{1}{\sqrt{2}}\right)}=\sqrt{2}$
$\sec 45^{\circ}=\frac{1}{x}=\frac{1}{\left(\frac{1}{\sqrt{2}}\right)}=\sqrt{2}$
$\cot 45^{\circ}=\frac{x}{y}=\frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}}=1$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Prove that the point A(1, 3, 0), B(-5, 5, 2), C(-9, -1, 2) and D(-3, -3, 0) taken in order are the vertices of a parallelogram. Also, show that ABCD is not a rectangle.
Find the equation of a straight line through the point of intersection of the lines 4x - 3y = 0 and 2x - 5y + 3 = 0 and parallel to 4x + 5y + 6 = 0.
Find the angles between the following pairs of straight lines:
$(m^2 - mn)y = (mn + n^2)x + n^3 and (mn + m^2)y = (mn - n^2)x + m^3.$
→$(m^2 - mn)y = (mn + n^2)x + n^3 and (mn + m^2)y = (mn - n^2)x + m^3.$
Two tangents to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ make angles $\theta_1, \theta_2$, with the transverse axis.
→Find the locus of their point of intersection if $\tan \theta_1+\tan \theta_2=k$.
There are three social media groups on a mobile: Group I, Group II and Group III. The
→probabilities that Group I, Group II and Group III sending the messages on sports are $\frac{2}{5}, \frac{1}{2}$
and $\frac{2}{3}$ respectively. The probability of opening the messages by Group I, Group II and
Group III are $\frac{1}{2}, \frac{1}{4}$ and $\frac{1}{4}$ respectively. Randomly one of the messages is opened and
found a message on sports. What is the probability that the message was from Group III.
Find the angles between the following pairs of straight lines:
3x + y + 12 = 0 and x + 2y - 1 = 0
3x + y + 12 = 0 and x + 2y - 1 = 0
Find the equation of the ellipse in standard form if : the distance between foci is 6 and the distance between directrices is $\frac{50}{3}$.
→Show that the line $3 x-4 y+10=0$ is a tangent to the hyperbola $x^2-4 y^2=20$. Also, findthe point of contact.
→Prove by the method of induction, for all n ∈ N.
→$1^2+4^2+7^2+\ldots \ldots+(3 n-2)^2=\frac{n}{2}\left(6 n^2-3 n-1\right)$
Calculate the mean, median and standard deviation of the following distribution:
→| Class-interval: | $31-35$ | $36-40$ | $41-45$ | $46-50$ | $51-55$ | $56-60$ | $61-65$ | $66-70$ |
| Frequency: | $2$ | $3$ | $8$ | $12$ | $16$ | $5$ | $2$ | $3$ |